?In the following exercises, evaluate the triple integrals \(\iiint_{E} f(x, y, z) d V\)

In the following exercises, evaluate the triple integrals \(\iiint_{E} f(x, y, z) d V\) over the solid E. \(f(x, y, z)=z\), \(B=\left\{(x, y, z) \mid x^{2}+y^{2} \leq 9, x \geq 0, y \geq 0,0 \leq z \leq 1\right\}\) Text Transcription: \iiint_{E} f(x, y, z) dV f(x, y, z)=z, B=\left\{(x, y, z) \mid x^{2}+y^{2} \leq 9, x \geq 0, y \geq 0,0 \leq z \leq 1\right\}

In the following exercises, evaluate the triple integrals \(\iiint_{E} f(x, y, z) d V\) over the solid E. \(f(x, y, z)=x z^{2}\), \(B=\left\{(x, y, z) \mid x^{2}+y^{2} \leq 16, x \geq 0, y \leq 0,-1 \leq z \leq 1\right\}\) Text Transcription: \iiint_{E} f(x, y, z) dV f(x, y, z)=x z^{2} B=\left\{(x, y, z) \mid x^{2}+y^{2} \leq 16, x \geq 0, y \leq 0,-1 \leq z \leq 1\right\}

In the following exercises, evaluate the triple integrals \(\iiint_{E} f(x, y, z) d V\) over the solid E. \(f(x, y, z)=x y\), \(B=\left\{(x, y, z) \mid x^{2}+y^{2} \leq 1, x \geq 0, x \geq y,-1 \leq z \leq 1\right\}\) Text Transcription: \iiint_{E} f(x, y, z) dV f(x, y, z)=x y B=\left\{(x, y, z) \mid x^{2}+y^{2} \leq 1, x \geq 0, x \geq y,-1 \leq z \leq 1\right\}

In the following exercises, evaluate the triple integrals \(\iiint_{E} f(x, y, z) d V\) over the solid E. \(f(x, y, z)=x^{2}+y^{2}\), \(B=\left\{(x, y, z) \mid x^{2}+y^{2} \leq 4, x \geq 0, x \leq y, 0 \leq z \leq 3\right\}\) Text Transcription: \iiint_{E} f(x, y, z) dV f(x, y, z)=x^{2}+y^{2} B=\left\{(x, y, z) \mid x^{2}+y^{2} \leq 4, x \geq 0, x \leq y, 0 \leq z \leq 3\right\}

In the following exercises, evaluate the triple integrals \(\iiint_{E} f(x, y, z) d V\) over the solid E. \(f(x, y, z)=e^{\sqrt{x^{2}+y^{2}}}\), \(B=\left\{(x, y, z) \mid 1 \leq x^{2}+y^{2} \leq 4, y \leq 0, x \leq y \sqrt{3}, 2 \leq z \leq 3\right\}\) Text Transcription: \iiint_{E} f(x, y, z) dV f(x, y, z)=e^{\sqrt{x^{2}+y^{2} B=\left\{(x, y, z) \mid 1 \leq x^{2}+y^{2} \leq 4, y \leq 0, x \leq y \sqrt{3}, 2 \leq z \leq 3\right\}

In the following exercises, evaluate the triple integrals \(\iiint_{E} f(x, y, z) d V\) over the solid E. \(f(x, y, z)=\sqrt{x^{2}+y^{2}}\), \(B=\left\{(x, y, z) \mid 1 \leq x^{2}+y^{2} \leq 9, y \leq 0,0 \leq z \leq 1\right\}\) Text Transcription: \iiint_{E} f(x, y, z) dV f(x, y, z)=\sqrt{x^{2}+y^{2} B=\left\{(x, y, z) \mid 1 \leq x^{2}+y^{2} \leq 9, y \leq 0,0 \leq z \leq 1\right\}

a. Let B be a cylindrical shell with inner radius a, outer radius b, and height c, where \(0<a<b\) and \(c>0\). Assume that a function F defined on B can be expressed in cylindrical coordinates as \(F(x, y, z)=f(r)+h(z)\), where f and h are differentiable functions. If \(\int_{a}^{b} \tilde{f}(r) d r=0\) and \(\tilde{h}(0)=0\), where \(\tilde{f}\) and \(\tilde{h}\) are antiderivatives of f and h, respectively, show that \(\iint_{B} F(x, y, z) d V=2 \pi c(b \tilde{f}(b)-a \tilde{f}(a))+\pi\left(b^{2}-a^{2}\right) \tilde{h}(c)\). b. Use the previous result to show that \(\iiint_{B}\left(z+\sin \sqrt{x^{2}}+y^{2}\right) d x \ d y \ d z=6 \pi^{2}(\pi-2)\), where B is a cylindrical shell with inner radius \(\pi\), outer radius \(2 \pi\), and height 2. Text Transcription: 0<a<b\ c>0\ F(x, y, z)=f(r)+h(z) \int_{a}^{b} \tilde{f}(r) d r=0 \tilde{h}(0)=0\ \tilde{f}\ \tilde{h}\ \iint_{B} F(x, y, z) d V=2 \pi c(b \tilde{f}(b)-a \tilde{f}(a))+\pi\left(b^{2}-a^{2}\right) \tilde{h}(c) \iiint_{B}\left(z+\sin \sqrt{x^{2}}+y^{2}\right) d x d y d z=6 \pi^{2}(\pi-2) \pi 2 \pi

a. Let B be a cylindrical shell with inner radius a, outer radius b, and height c, where 0 < a < b and c > 0. Assume that a function F defined on B can be expressed in cylindrical coordinates as \(F(x, y, z)=f(r) g(\theta) h(z)\), where f, g, and h are differential functions. If \(\int_{a}^{b} \tilde{f}(r) d r=0\), where \(\tilde{f}\) is an antiderivative of f, show that \(\iint_{B} F(x, y, z) d V=[b \tilde{f}(b)-a \tilde{f}(a)][\tilde{g}(2 \pi)-\tilde{g}(0)][\tilde{h}(c)-\tilde{h}(0)]\) where \(\tilde{g}\) and \(\tilde{h}\) are antiderivatives of g and h, respectively. b. Use the previous result to show that \(\iiint_{B} z \sin \sqrt{x^{2}+y^{2}} d x \ d y \ d z=-12 \pi^{2}\), where B is a cylindrical shell with inner radius \(\pi\), outer radius \(2 \pi\), and height 2. Text Transcription: F(x, y, z)=f(r) g(\theta) h(z) \int_{a}^{b} \tilde{f}(r) d r=0 \tilde{f} \iint_{B} F(x, y, z) d V=[b \tilde{f}(b)-a \tilde{f}(a)][\tilde{g}(2 \pi)-\tilde{g}(0)][\tilde{h}(c)-\tilde{h}(0)] \tilde{g} \tilde{h} \iiint_{B} z \sin \sqrt{x^{2}+y^{2}} d x d y d z=-12 \pi^{2} B \pi 2 \pi

In the following exercises, the boundaries of the solid E are given in cylindrical coordinates. a. Express the region E in cylindrical coordinates. b. Convert the integral \(\iiint_{E} f(x, y, z) d V\) to cylindrical coordinates. E is bounded by the right circular cylinder \(r=4 \sin \theta\), the \(r \theta\)-plane and the sphere \(r^{2}+z^{2}=16\). Text Transcription: \iiint_{E} f(x, y, z) dV r=4 \sin \theta r \theta r^{2}+z^{2}=16

In the following exercises, the boundaries of the solid E are given in cylindrical coordinates. a. Express the region E in cylindrical coordinates. b. Convert the integral \(\iiint_{E} f(x, y, z) d V\) to cylindrical coordinates. E is bounded by the right circular cylinder \(r=\cos \theta\), the \(r \theta\)-plane and the sphere \(r^{2}+z^{2}=9\). Text Transcription: \iiint_{E} f(x, y, z) d V \r=\cos \theta r \theta r^{2}+z^{2}=9

In the following exercises, the boundaries of the solid E are given in cylindrical coordinates. a. Express the region E in cylindrical coordinates. b. Convert the integral \(\iiint_{E} f(x, y, z) d V\) to cylindrical coordinates. E is located in the first octant and is bounded by the circular paraboloid \(z=9-3 r^{2}\), the cylinder \(r=\sqrt{3}\), and the plane \(r(\cos \theta+\sin \theta)=20-z\). Text Transcription: \iiint_{E} f(x, y, z) dV z=9-3 r^{2} r=\sqrt{3} r(\cos \theta+\sin \theta)=20-z

In the following exercises, the boundaries of the solid E are given in cylindrical coordinates. a. Express the region E in cylindrical coordinates. b. Convert the integral \(\iiint_{E} f(x, y, z) d V\) to cylindrical coordinates. E is located in the first octant outside the circular paraboloid \(z=10-2 r^{2}\) and inside the cylinder \(r=\sqrt{5}\) and is bounded also by the planes \(z=20\) and \(\theta=\frac{\pi}{4}\). Text Transcription: \iiint_{E} f(x, y, z) dV z=10-2 r^{2} r=\sqrt{5} z=20 \theta=\frac{\pi}{4}

In the following exercises, the function f and region E are given. a. Express the region E and the function f in cylindrical coordinates. b. Convert the integral \(\iiint_{B} f(x, y, z) d V\) into cylindrical coordinates and evaluate it. \(f(x, y, z)=\frac{1}{x+3}\), \(E=\left\{(x, y, z) \mid 0 \leq x^{2}+y^{2} \leq 9, x \geq 0, y \geq 0,0 \leq z \leq x+3\right\}\) Text Transcription: \iiint_{B} f(x, y, z) dV f(x, y, z)=\frac{1}{x+3} E=\left\{(x, y, z) \mid 0 \leq x^{2}+y^{2} \leq 9, x \geq 0, y \geq 0,0 \leq z \leq x+3\right\}

In the following exercises, the function f and region E are given. a. Express the region E and the function f in cylindrical coordinates. b. Convert the integral \(\iiint_{B} f(x, y, z) d V\) into cylindrical coordinates and evaluate it. \(f(x, y, z)=x^{2}+y^{2}\), \(E=\left\{(x, y, z) \mid 0 \leq x^{2}+y^{2} \leq 4, y \geq 0,0 \leq z \leq 3-x\right\}\) Text Transcription: \iiint_{B} f(x, y, z) d V f(x, y, z)=x^{2}+y^{2} E=\left\{(x, y, z) \mid 0 \leq x^{2}+y^{2} \leq 4, y \geq 0,0 \leq z \leq 3-x\right\}

In the following exercises, the function f and region E are given. a. Express the region E and the function f in cylindrical coordinates. b. Convert the integral \(\iiint_{B} f(x, y, z) d V\) into cylindrical coordinates and evaluate it. \(f(x, y, z)=x\), \(E=\left\{(x, y, z) \mid 1 \leq y^{2}+z^{2} \leq 9,0 \leq x \leq 1-y^{2}-z^{2}\right\}\) Text Transcription: \iiint_{B} f(x, y, z) dV f(x, y, z)=x E=\left\{(x, y, z) \mid 1 \leq y^{2}+z^{2} \leq 9,0 \leq x \leq 1-y^{2}-z^{2}\right\}

In the following exercises, the function f and region E) a. Express the region E and the function f in cylindrical coordinates. b. Convert the integral \(\iiint_{B} f(x, y, z) d V\) into cylindrical coordinates and evaluate it. \(f(x, y, z)=y\), \(E=\left\{(x, y, z) \mid 1 \leq x^{2}+z^{2} \leq 9,0 \leq y \leq 1-x^{2}-z^{2}\right\}\) Text Transcription: \iiint_{B} f(x, y, z) d V f(x, y, z)=y E=\left\{(x, y, z) \mid 1 \leq x^{2}+z^{2} \leq 9,0 \leq y \leq 1-x^{2}-z^{2}\right\}

In the following exercises, find the volume of the solid E whose boundaries are given in rectangular coordinates. E is above the xy-plane, inside the cylinder \(x^{2}+y^{2}=1\), and below the plane z=1. Text transcription: x^{2}+y^{2}=1

In the following exercises, find the volume of the solid E whose boundaries are given in rectangular coordinates. E is below the plane z=1 and inside the paraboloid \(z=x^{2}+y^{2}\). Text Transcription: z=x^{2}+y^{2}

In the following exercises, find the volume of the solid E whose boundaries are given in rectangular coordinates. E is bounded by the circular cone \(z=\sqrt{x^{2}+y^{2}}\) and z=1. Text Transcription: (z=\sqrt{x^{2}+y^{2}}

In the following exercises, find the volume of the solid E whose boundaries are given in rectangular coordinates. E is located above the xy-plane, below z=1, outside the one-sheeted hyperboloid \(x^{2}+y^{2}-z^{2}=1\), and inside the cylinder \(x^{2}+y^{2}=2\). Text Transcription: x^{2}+y^{2}-z^{2}=1 x^{2}+y^{2}=2

In the following exercises, find the volume of the solid E whose boundaries are given in rectangular coordinates. E is located inside the cylinder \(x^{2}+y^{2}=1\) and between the circular paraboloids \(z=1-x^{2}-y^{2}\) and \(z=x^{2}+y^{2}\) Text Transcription: x^{2}+y^{2}=1 z=1-x^{2}-y^{2} z=x^{2}+y^{2}

In the following exercises, find the volume of the solid E whose boundaries are given in rectangular coordinates. E is located inside the sphere \(x^{2}+y^{2}+z^{2}=1\), above the xy-plane, and inside the circular cone \(z=\sqrt{x^{2}+y^{2}}\). Text Transcription: x^{2}+y^{2}+z^{2}=1 z=\sqrt{x^{2}+y^{2}}

In the following exercises, find the volume of the solid E whose boundaries are given in rectangular coordinates. E is located outside the circular cone \(x^{2}+y^{2}=(z-1)^{2}\) and between the planes z=0 and z=2. Text Transcription: x^{2}+y^{2}=(z-1)^{2}

In the following exercises, find the volume of the solid E whose boundaries are given in rectangular coordinates. E is located outside the circular cone \(z=1-\sqrt{x^{2}+y^{2}}\), above the xy-plane, below the circular paraboloid, and between the planes z=0 and z=2. Text Transcription: z=1-\sqrt{x^{2}+y^{2}}

In the following exercises, find the volume of the solid E whose boundaries are given in rectangular coordinates. [T] Use a computer algebra system (CAS) to graph the solid whose volume is given by the iterated integral in cylindrical coordinates r dz dr d\(\theta\). Find the volume V of the solid. Round your answer to four decimal places. Text Transcription: theta

In the following exercises, find the volume of the solid E whose boundaries are given in rectangular coordinates. [T] Use a CAS to graph the solid whose volume is given by the iterated integral in cylindrical coordinates r dz dr d\(\theta\). Find the volume V of the solid Round your answer to four decimal places. Text Transcription: theta

In the following exercises, find the volume of the solid E whose boundaries are given in rectangular coordinates. Convert the integral xz dz dx dy into an integral in cylindrical coordinates.

In the following exercises, find the volume of the solid E whose boundaries are given in rectangular coordinates. Convert the integral (xy + z)dz dx dy into an integral in cylindrical coordinates.

In the following exercises, evaluate the triple integral f (x, y, z)dV over the solid B. f (x, y, z) = 1, B = {(x, y, z)|x^2 + y^2 + z^2 \(\leq\) 90, z \(\leq\) 0} Text Transcription: \leq

In the following exercises, evaluate the triple integral f (x, y, z)dV over the solid B. f(x, y, z) = 1 -\(\sqrt{}\)x^2 + y^2 + z^2, B = {(x, y, z)|x^2 + y^2 + z^2 \(\leq\) 9, y \(\geq\) 0, z \(\geq\) 0 Text Transcription: geq sqrt

In the following exercises, evaluate the triple integral f (x, y, z)dV over the solid B. f(x, y, z) = \(\sqrt{}\)x^2 + y^2, B is bounded above by the half-sphere x2 + y2 + z2 = 9 with z \(\geq\) 0 Text Transcription: sqrt geq

In the following exercises, evaluate the triple integral f (x, y, z)dV over the solid B. f(x, y, z) = z, B is bounded above by the half-sphere x^2 + y^2 + z^2 = 16 with z \(\geq\) 0 and below by the cone 2z^2 = x^2 + y^2. Text Transcription: geq

In the following exercises, evaluate the triple integral f (x, y, z)dV over the solid B. Show that if \(F(\rho, \theta, \varphi)=f(\rho) g(\theta) h(\varphi)\) is a continuous function on the spherical box \(B=\{(\rho, \theta, \varphi) \mid a \leq \rho \leq b, \alpha \leq \theta \leq \beta, \gamma \leq \varphi \leq \psi\}\), the \(\iiint_{B} F d V=\left(\int_{a}^{b} \rho^{2} f(\rho) d r\right)\left(\int_{\alpha}^{\beta} g(\theta) d \theta\right)\left(\int_{\gamma}^{\psi} h(\varphi) \sin \varphi d \varphi\right)\) Text Transcription: F(rho,theta,varphi)=f(rho) g(theta) h(varphi) B=(rho,theta,varphi)mid_a_leq_rho_leq_b,alpha_leq_theta_leq_beta,gamma_leq_varphi_leq _psi} iiint_B_F_d_V=left(int_a^b_rho^2_f(rho)_d_r.right)left(int_alpha^beta_g(theta)d_theta\right)\left(\int_gamma^psi_h(varphi)sin_varphi_d_varphi_right)

In the following exercises, find the volume of the solid E whose boundaries are given in rectangular coordinates.

a. Let B be the region between the upper concentric hemispheres of radii a and b centered at the origin and situated in the first octant, where 0 < a < b. Consider F a function defined on B whose form in spherical coordinates \((\rho, \theta, \varphi)\) is \(F(x, y, z)=f(\rho) \cos \varphi\). Show that if g(a) = g(b) = 0 and \(\int_{a}^{b} h(\rho) d \rho=0\), then \(\iiint_{B} F(x, y, z) d V=\frac{\pi^{2}}{4}[a h(a)-b h(b)]\), where g is an antiderivative of f and h is an antiderivative of g. b. Use the previous result to show that \(\iiint_{B} \frac{z \cos \sqrt{x^{2}+y^{2}+z^{2}}}{\sqrt{x^{2}+y^{2}+z^{2}}} d V=\frac{3 \pi^{2}}{2}\) , where B is the region between the upper concentric hemispheres of radii \(\pi\) and \(2 \pi\) centered at the origin and situated in the first octant. Text Transcription: (rho, theta, varphi) F(x, y, z) = f(rho) cos varphi int_a^b h(rho) d rho = 0 iiint_B F(x, y, z) dV = pi^2 / 4 [a h(a)-b h(b)] iiint_B z cos sqrt x^2 + y^2 + z^2 / sqrt x^2 + y^2 + z^2 dV =3 pi^2 / 2 pi 2pi

In the following exercises, the function f and region E are given. a. Express the region E and function f in cylindrical coordinates. b. Convert the integral \(\iiint_{B} f(x, y, z) d V\) into cylindrical coordinates and evaluate it. f(x, y, z) = z; \(E=\left\{(x, y, z) \mid 0 \leq x^{2}+y^{2}+z^{2} \leq 1, z \geq 0\right\}\) Text Transcription: iiint_B f(x, y, z) dV E = {(x, y, z) mid 0 leq x^2 + y^2 + z^2 leq 1, z geq 0}

In the following exercises, the function f and region E are given. a. Express the region E and function f in cylindrical coordinates. b. Convert the integral \(\iiint_{B} f(x, y, z) d V\) into cylindrical coordinates and evaluate it. f(x, y, z) = x + y; \(E=\left\{(x, y, z) \mid 1 \leq x^{2}+y^{2}+z^{2} \leq 2, z \geq 0, y \geq 0\right\}\) Text Transcription: iiint_B f(x, y, z) dV E = {(x, y, z) mid 1 leq x^2 + y^2 + z^2 leq 2, z geq 0, y geq 0}

In the following exercises, the function f and region E are given. a. Express the region E and function f in cylindrical coordinates. b. Convert the integral \(\iiint_{B} f(x, y, z) d V\) into cylindrical coordinates and evaluate it. f(x, y, z) = 2xy; \(E=\left\{(x, y, z) \mid \sqrt{x^{2}+y^{2}} \leq z \leq \sqrt{1-x^{2}-y^{2}}, x \geq 0, y \geq 0\right\}\) Text Transcription: iiint_B f(x, y, z) dV E = {(x, y, z) mid sqrt x^2 + y^2 leq z leq sqrt 1 - x^2 - y^2, x geq 0, y geq 0}

In the following exercises, the function f and region E are given. a. Express the region E and function f in cylindrical coordinates. b. Convert the integral \(\iiint_{B} f(x, y, z) d V\) into cylindrical coordinates and evaluate it. f(x, y, z) = z; \(E=\left\{(x, y, z) \mid x^{2}+y^{2}+z^{2}-2 z \leq 0, \sqrt{x^{2}+y^{2}} \leq z\right\}\) Text Transcription: iiint_B f(x, y, z) dV E = {(x, y, z) mid x^2 + y^2 + z^2 - 2z leq 0, sqrt x^2 + y^2 leq z}

In the following exercises, find the volume of the solid E whose boundaries are given in rectangular coordinates. \(E=\left\{(x, y, z) \mid \sqrt{x^{2}+y^{2}} \leq z \leq \sqrt{16-x^{2}-y^{2}}, x \geq 0, y \geq 0\right\}\) Text Transcription: E = {(x, y, z) mid sqrt x^2 + y^2 leq z leq sqrt 16 - x^2 - y^2, x geq 0, y geq 0}

In the following exercises, find the volume of the solid E whose boundaries are given in rectangular coordinates. \(E=\left\{(x, y, z) \mid x^{2}+y^{2}+z^{2}-2 z \leq 0, \sqrt{x^{2}+y^{2}} \leq z\right\}\) Text Transcription: E = {(x, y, z) mid x^2 + y^2 + z^2 - 2z leq 0, sqrt x^2 + y^2 leq z}

Use spherical coordinates to find the volume of the solid situated outside the sphere \(\rho=1\) and inside the sphere \(\rho=\cos \varphi\), with \(\varphi \in\left[0, \frac{\pi}{2}\right]\) . Text Transcription: rho=1 rho=cos varphi varphi in [0, pi/2]

Use spherical coordinates to find the volume of the ball \(\rho \leq 3\) that is situated between the cones \(\varphi=\frac{\pi}{4}\) and \(\varphi=\frac{\pi}{3}\) . Text Transcription: rho leq 3 varphi=pi/4 varphi=pi/3

Convert the \(\int_{-4}^{4} \int_{-\sqrt{16-y^{2}}}^{\sqrt{16-y^{2}}} \int_{-\sqrt{16-x^{2}-y^{2}}}^{\sqrt{16-x^{2}-y^{2}}} \left(x^{2}+y^{2}+z^{2}\right) d z d x d y\) into an integral in spherical coordinates. Text Transcription: int_-4^4 int_-sqrt 16-y^2 ^ sqrt 16-y^2 int_-sqrt 16-x^2-y^2 ^ sqrt 16-x^2-y^2 (x^2+y^2+z^2) dzdxdy

Convert the \(\int_{0}^{4} \int_{0}^{\sqrt{16-x^{2}}} \int_{-\sqrt{16-x^{2}-y^{2}}}^{\sqrt{16-x^{2}}}\left(x^{2}+y^{2}+z^{2}\right)^{2} d z d y d x) into an integral in spherical coordinates. Text Transcription: int_0^4 int_0 ^ sqrt 16-x^2 int_-sqrt 16-x^2-y^2 ^ sqrt 16-x^2 (x^2+y^2+z^2)^2 dzdydx

Convert the integral \(\int_{-2}^{2} \int_{-\sqrt{4-x^{2}}}^{\sqrt{4-x^{2}}} \int_{ \sqrt{x^{2}+y^{2}}}^{\sqrt{16-x^{2}-y^{2}}} d z d y d x\) into an integral in spherical coordinates and evaluate it. Text Transcription: int_-2 ^ 2 int_-sqrt 4-x^2 ^ sqrt 4-x^2 int_sqrt x^2+y^2 ^ sqrt 16-x^2-y^2 dzdydx

[T] Use a CAS to graph the solid whose volume is given by the iterated integral in spherical coordinates \(\int_{\pi / 2}^{\pi} \int_{5 \pi / 6}^{\pi / 6} \int_{0}^{2} \rho^{2} \sin \varphi d \rho d \varphi d \theta\). Find the volume V of the solid. Round your answer to three decimal places. Text Transcription: int_pi/2 ^ pi int_5 pi/6 ^ pi / 6 int_0 ^ 2 rho^2 sin varphi d rho d varphi d theta

[T] Use a CAS to graph the solid whose volume is given by the iterated integral in spherical coordinates as \(\int_{0}^{2 \pi} \int_{3 \pi / 4}^{\pi / 4} \int_{0}^{1} \rho^{2} \sin \varphi d \rho d \varphi d \theta\). Find the volume V of the solid. Round your answer to three decimal places. Text Transcription: int_0 ^ 2 pi int_3 pi/4 ^ pi/4 int_0 ^ 1 rho^2 sin varphi d rho d varphi d theta

[T] Use a CAS to evaluate the integral \(\iiint_{E}\left(x^{2}+y^{2}\right) d V\) where E lies above the paraboloid \(z=x^{2}+y^{2}\) and below the plane z = 3y. Text Transcription: iiint_E (x^2+y^2) dV z=x^2+y^2

[T] a. Evaluate the integral \(\iiint_{E} e^{\sqrt{x^{2}+y^{2}+z^{2}}} d V\), where E is bounded by the spheres \(4 x^{2}+4 y^{2}+4 z^{2}=1\) and \(x^{2}+y^{2}+z^{2}=1\). b. Use a CAS to find an approximation of the previous integral. Round your answer to two decimal places. Text Transcription: iiint_E e^sqrt x^2+y^2+z^2 dV 4x^2+4y^2+4z^2=1 x^2+y^2+z^2=1

Express the volume of the solid inside the sphere \(x^{2}+y^{2}+z^{2}=16\) and outside the cylinder \(x^{2}+y^{2}=4\) as triple integrals in cylindrical coordinates and spherical coordinates, respectively. Text Transcription: x^2+y^2+z^2=16 x^2+y^2=4

Express the volume of the solid inside the sphere \(x^{2}+y^{2}+z^{2}=16\) and outside the cylinder \(x^{2}+y^{2}=4\) that is located in the first octant as triple integrals in cylindrical coordinates and spherical coordinates, respectively. Text Transcription: x^2+y^2+z^2=16 x^2+y^2=4

The power emitted by an antenna has a power density per unit volume given in spherical coordinates by \(p(\rho, \theta, \varphi)=\frac{P_{0}}{\rho^{2}} \cos^{2} \theta \sin^{4} \varphi\), where \(P_{0}\) is a constant with units in watts. The total power within a sphere B of radius r meters is defined as \(P=\iiint_{B} p(\rho, \theta, \varphi) d V\). Find the total power P. Text Transcription: p(rho, theta, varphi)=P_0 / rho^2 cos^2 theta sin^4 varphi P_0 P=iiint_B p(rho, theta, varphi) dV

Use the preceding exercise to find the total power within a sphere B of radius 5 meters when the power density per unit volume is given by \(p(\rho, \theta, \varphi)=\frac{30}{\rho^{2}} \cos ^{2} \theta \sin ^{4} \varphi\) . Text Transcription: p(rho, theta, varphi)=frac30 / rho^2 cos^2 theta sin^4 varphi

A charge cloud contained in a sphere B of radius r centimeters centered at the origin has its charge density given by \(q(x, y, z)=k \sqrt{x^{2}+y^{2}+z^{2}} \frac{\mu \mathrm{C}}{\mathrm{cm}^{3}}\) , where k > 0 . The total charge contained in B is given by \(Q=\iiint_{R} q(x, y, z) d V\) . Find the total charge Q . Text Transcription: q(x, y, z)=k sqrt x^2+y^2+z^2 mu C / cm^3 Q=iiint_R q(x, y, z) dV

Use the preceding exercise to find the total charge cloud contained in the unit sphere if the charge density is \(q(x, y, z)=20 \sqrt{x^{2}+y^{2}+z^{2}} \frac{\mu \mathrm{C}}{\mathrm{cm}^{3}} .\) . Text Transcription: q(x, y, z)=20 sqrt x^2+y^2+z^2 mu C / cm^3

In the following exercises, find the volume of the solid E whose boundaries are given in rectangular coordinates.

a. Let B be the region between the upper concentric hemispheres of radii a and b centered at the origin and situated in the first octant, where 0 < a < b. Consider F a function defined on B whose form in spherical coordinates \((\rho, \theta, \varphi)\) is \(F(x, y, z)=f(\rho) \cos \varphi\). Show that if g(a) = g(b) = 0 and \(\int_{a}^{b} h(\rho) d \rho=0\), then \(\iiint_{B} F(x, y, z) d V=\frac{\pi^{2}}{4}[a h(a)-b h(b)]\), where g is an antiderivative of f and h is an antiderivative of g. b. Use the previous result to show that \(\iiint_{B} \frac{z \cos \sqrt{x^{2}+y^{2}+z^{2}}}{\sqrt{x^{2}+y^{2}+z^{2}}} d V=\frac{3 \pi^{2}}{2}\) , where B is the region between the upper concentric hemispheres of radii \(\pi\) and \(2 \pi\) centered at the origin and situated in the first octant. Text Transcription: (rho, theta, varphi) F(x, y, z) = f(rho) cos varphi int_a^b h(rho) d rho = 0 iiint_B F(x, y, z) dV = pi^2 / 4 [a h(a)-b h(b)] iiint_B z cos sqrt x^2 + y^2 + z^2 / sqrt x^2 + y^2 + z^2 dV =3 pi^2 / 2 pi 2pi

In the following exercises, the function f and region E are given. a. Express the region E and function f in cylindrical coordinates. b. Convert the integral \(\iiint_{B} f(x, y, z) d V\) into cylindrical coordinates and evaluate it. f(x, y, z) = z; \(E=\left\{(x, y, z) \mid 0 \leq x^{2}+y^{2}+z^{2} \leq 1, z \geq 0\right\}\) Text Transcription: iiint_B f(x, y, z) dV E = {(x, y, z) mid 0 leq x^2 + y^2 + z^2 leq 1, z geq 0}

In the following exercises, the function f and region E are given. a. Express the region E and function f in cylindrical coordinates. b. Convert the integral \(\iiint_{B} f(x, y, z) d V\) into cylindrical coordinates and evaluate it. f(x, y, z) = x + y; \(E=\left\{(x, y, z) \mid 1 \leq x^{2}+y^{2}+z^{2} \leq 2, z \geq 0, y \geq 0\right\}\) Text Transcription: iiint_B f(x, y, z) dV E = {(x, y, z) mid 1 leq x^2 + y^2 + z^2 leq 2, z geq 0, y geq 0}

In the following exercises, the function f and region E are given. a. Express the region E and function f in cylindrical coordinates. b. Convert the integral \(\iiint_{B} f(x, y, z) d V\) into cylindrical coordinates and evaluate it. f(x, y, z) = 2xy; \(E=\left\{(x, y, z) \mid \sqrt{x^{2}+y^{2}} \leq z \leq \sqrt{1-x^{2}-y^{2}}, x \geq 0, y \geq 0\right\}\) Text Transcription: iiint_B f(x, y, z) dV E = {(x, y, z) mid sqrt x^2 + y^2 leq z leq sqrt 1 - x^2 - y^2, x geq 0, y geq 0}

In the following exercises, the function f and region E are given. a. Express the region E and function f in cylindrical coordinates. b. Convert the integral \(\iiint_{B} f(x, y, z) d V\) into cylindrical coordinates and evaluate it. f(x, y, z) = z; \(E=\left\{(x, y, z) \mid x^{2}+y^{2}+z^{2}-2 z \leq 0, \sqrt{x^{2}+y^{2}} \leq z\right\}\) Text Transcription: iiint_B f(x, y, z) dV E = {(x, y, z) mid x^2 + y^2 + z^2 - 2z leq 0, sqrt x^2 + y^2 leq z}

In the following exercises, find the volume of the solid E whose boundaries are given in rectangular coordinates. \(E=\left\{(x, y, z) \mid \sqrt{x^{2}+y^{2}} \leq z \leq \sqrt{16-x^{2}-y^{2}}, x \geq 0, y \geq 0\right\}\) Text Transcription: E = {(x, y, z) mid sqrt x^2 + y^2 leq z leq sqrt 16 - x^2 - y^2, x geq 0, y geq 0}

In the following exercises, find the volume of the solid E whose boundaries are given in rectangular coordinates. \(E=\left\{(x, y, z) \mid x^{2}+y^{2}+z^{2}-2 z \leq 0, \sqrt{x^{2}+y^{2}} \leq z\right\}\) Text Transcription: E = {(x, y, z) mid x^2 + y^2 + z^2 - 2z leq 0, sqrt x^2 + y^2 leq z}

Use spherical coordinates to find the volume of the solid situated outside the sphere \(\rho=1\) and inside the sphere \(\rho=\cos \varphi\), with \(\varphi \in\left[0, \frac{\pi}{2}\right]\) . Text Transcription: rho=1 rho=cos varphi varphi in [0, pi/2]

Use spherical coordinates to find the volume of the ball \(\rho \leq 3\) that is situated between the cones \(\varphi=\frac{\pi}{4}\) and \(\varphi=\frac{\pi}{3}\) . Text Transcription: rho leq 3 varphi=pi/4 varphi=pi/3

Convert the \(\int_{-4}^{4} \int_{-\sqrt{16-y^{2}}}^{\sqrt{16-y^{2}}} \int_{-\sqrt{16-x^{2}-y^{2}}}^{\sqrt{16-x^{2}-y^{2}}} \left(x^{2}+y^{2}+z^{2}\right) d z d x d y\) into an integral in spherical coordinates. Text Transcription: int_-4^4 int_-sqrt 16-y^2 ^ sqrt 16-y^2 int_-sqrt 16-x^2-y^2 ^ sqrt 16-x^2-y^2 (x^2+y^2+z^2) dzdxdy

Convert the \(\int_{0}^{4} \int_{0}^{\sqrt{16-x^{2}}} \int_{-\sqrt{16-x^{2}-y^{2}}}^{\sqrt{16-x^{2}}}\left(x^{2}+y^{2}+z^{2}\right)^{2} d z d y d x) into an integral in spherical coordinates. Text Transcription: int_0^4 int_0 ^ sqrt 16-x^2 int_-sqrt 16-x^2-y^2 ^ sqrt 16-x^2 (x^2+y^2+z^2)^2 dzdydx

Convert the integral \(\int_{-2}^{2} \int_{-\sqrt{4-x^{2}}}^{\sqrt{4-x^{2}}} \int_{ \sqrt{x^{2}+y^{2}}}^{\sqrt{16-x^{2}-y^{2}}} d z d y d x\) into an integral in spherical coordinates and evaluate it. Text Transcription: int_-2 ^ 2 int_-sqrt 4-x^2 ^ sqrt 4-x^2 int_sqrt x^2+y^2 ^ sqrt 16-x^2-y^2 dzdydx

[T] Use a CAS to graph the solid whose volume is given by the iterated integral in spherical coordinates \(\int_{\pi / 2}^{\pi} \int_{5 \pi / 6}^{\pi / 6} \int_{0}^{2} \rho^{2} \sin \varphi d \rho d \varphi d \theta\). Find the volume V of the solid. Round your answer to three decimal places. Text Transcription: int_pi/2 ^ pi int_5 pi/6 ^ pi / 6 int_0 ^ 2 rho^2 sin varphi d rho d varphi d theta

[T] Use a CAS to graph the solid whose volume is given by the iterated integral in spherical coordinates as \(\int_{0}^{2 \pi} \int_{3 \pi / 4}^{\pi / 4} \int_{0}^{1} \rho^{2} \sin \varphi d \rho d \varphi d \theta\). Find the volume V of the solid. Round your answer to three decimal places. Text Transcription: int_0 ^ 2 pi int_3 pi/4 ^ pi/4 int_0 ^ 1 rho^2 sin varphi d rho d varphi d theta

[T] Use a CAS to evaluate the integral \(\iiint_{E}\left(x^{2}+y^{2}\right) d V\) where E lies above the paraboloid \(z=x^{2}+y^{2}\) and below the plane z = 3y. Text Transcription: iiint_E (x^2+y^2) dV z=x^2+y^2

[T] a. Evaluate the integral \(\iiint_{E} e^{\sqrt{x^{2}+y^{2}+z^{2}}} d V\), where E is bounded by the spheres \(4 x^{2}+4 y^{2}+4 z^{2}=1\) and \(x^{2}+y^{2}+z^{2}=1\). b. Use a CAS to find an approximation of the previous integral. Round your answer to two decimal places. Text Transcription: iiint_E e^sqrt x^2+y^2+z^2 dV 4x^2+4y^2+4z^2=1 x^2+y^2+z^2=1

Express the volume of the solid inside the sphere \(x^{2}+y^{2}+z^{2}=16\) and outside the cylinder \(x^{2}+y^{2}=4\) as triple integrals in cylindrical coordinates and spherical coordinates, respectively. Text Transcription: x^2+y^2+z^2=16 x^2+y^2=4

Express the volume of the solid inside the sphere \(x^{2}+y^{2}+z^{2}=16\) and outside the cylinder \(x^{2}+y^{2}=4\) that is located in the first octant as triple integrals in cylindrical coordinates and spherical coordinates, respectively. Text Transcription: x^2+y^2+z^2=16 x^2+y^2=4

The power emitted by an antenna has a power density per unit volume given in spherical coordinates by \(p(\rho, \theta, \varphi)=\frac{P_{0}}{\rho^{2}} \cos^{2} \theta \sin^{4} \varphi\), where \(P_{0}\) is a constant with units in watts. The total power within a sphere B of radius r meters is defined as \(P=\iiint_{B} p(\rho, \theta, \varphi) d V\). Find the total power P. Text Transcription: p(rho, theta, varphi)=P_0 / rho^2 cos^2 theta sin^4 varphi P_0 P=iiint_B p(rho, theta, varphi) dV

Use the preceding exercise to find the total power within a sphere B of radius 5 meters when the power density per unit volume is given by \(p(\rho, \theta, \varphi)=\frac{30}{\rho^{2}} \cos ^{2} \theta \sin ^{4} \varphi\) . Text Transcription: p(rho, theta, varphi)=frac30 / rho^2 cos^2 theta sin^4 varphi

A charge cloud contained in a sphere B of radius r centimeters centered at the origin has its charge density given by \(q(x, y, z)=k \sqrt{x^{2}+y^{2}+z^{2}} \frac{\mu \mathrm{C}}{\mathrm{cm}^{3}}\) , where k > 0 . The total charge contained in B is given by \(Q=\iiint_{R} q(x, y, z) d V\) . Find the total charge Q . Text Transcription: q(x, y, z)=k sqrt x^2+y^2+z^2 mu C / cm^3 Q=iiint_R q(x, y, z) dV

Use the preceding exercise to find the total charge cloud contained in the unit sphere if the charge density is \(q(x, y, z)=20 \sqrt{x^{2}+y^{2}+z^{2}} \frac{\mu \mathrm{C}}{\mathrm{cm}^{3}} .\) . Text Transcription: q(x, y, z)=20 sqrt x^2+y^2+z^2 mu C / cm^3

In the following exercises, find the volume of the solid E whose boundaries are given in rectangular coordinates.

a. Let B be the region between the upper concentric hemispheres of radii a and b centered at the origin and situated in the first octant, where 0 < a < b. Consider F a function defined on B whose form in spherical coordinates \((\rho, \theta, \varphi)\) is \(F(x, y, z)=f(\rho) \cos \varphi\). Show that if g(a) = g(b) = 0 and \(\int_{a}^{b} h(\rho) d \rho=0\), then \(\iiint_{B} F(x, y, z) d V=\frac{\pi^{2}}{4}[a h(a)-b h(b)]\), where g is an antiderivative of f and h is an antiderivative of g. b. Use the previous result to show that \(\iiint_{B} \frac{z \cos \sqrt{x^{2}+y^{2}+z^{2}}}{\sqrt{x^{2}+y^{2}+z^{2}}} d V=\frac{3 \pi^{2}}{2}\) , where B is the region between the upper concentric hemispheres of radii \(\pi\) and \(2 \pi\) centered at the origin and situated in the first octant. Text Transcription: (rho, theta, varphi) F(x, y, z) = f(rho) cos varphi int_a^b h(rho) d rho = 0 iiint_B F(x, y, z) dV = pi^2 / 4 [a h(a)-b h(b)] iiint_B z cos sqrt x^2 + y^2 + z^2 / sqrt x^2 + y^2 + z^2 dV =3 pi^2 / 2 pi 2pi

In the following exercises, the function f and region E are given. a. Express the region E and function f in cylindrical coordinates. b. Convert the integral \(\iiint_{B} f(x, y, z) d V\) into cylindrical coordinates and evaluate it. f(x, y, z) = z; \(E=\left\{(x, y, z) \mid 0 \leq x^{2}+y^{2}+z^{2} \leq 1, z \geq 0\right\}\) Text Transcription: iiint_B f(x, y, z) dV E = {(x, y, z) mid 0 leq x^2 + y^2 + z^2 leq 1, z geq 0}

In the following exercises, the function f and region E are given. a. Express the region E and function f in cylindrical coordinates. b. Convert the integral \(\iiint_{B} f(x, y, z) d V\) into cylindrical coordinates and evaluate it. f(x, y, z) = x + y; \(E=\left\{(x, y, z) \mid 1 \leq x^{2}+y^{2}+z^{2} \leq 2, z \geq 0, y \geq 0\right\}\) Text Transcription: iiint_B f(x, y, z) dV E = {(x, y, z) mid 1 leq x^2 + y^2 + z^2 leq 2, z geq 0, y geq 0}

In the following exercises, the function f and region E are given. a. Express the region E and function f in cylindrical coordinates. b. Convert the integral \(\iiint_{B} f(x, y, z) d V\) into cylindrical coordinates and evaluate it. f(x, y, z) = 2xy; \(E=\left\{(x, y, z) \mid \sqrt{x^{2}+y^{2}} \leq z \leq \sqrt{1-x^{2}-y^{2}}, x \geq 0, y \geq 0\right\}\) Text Transcription: iiint_B f(x, y, z) dV E = {(x, y, z) mid sqrt x^2 + y^2 leq z leq sqrt 1 - x^2 - y^2, x geq 0, y geq 0}

In the following exercises, the function f and region E are given. a. Express the region E and function f in cylindrical coordinates. b. Convert the integral \(\iiint_{B} f(x, y, z) d V\) into cylindrical coordinates and evaluate it. f(x, y, z) = z; \(E=\left\{(x, y, z) \mid x^{2}+y^{2}+z^{2}-2 z \leq 0, \sqrt{x^{2}+y^{2}} \leq z\right\}\) Text Transcription: iiint_B f(x, y, z) dV E = {(x, y, z) mid x^2 + y^2 + z^2 - 2z leq 0, sqrt x^2 + y^2 leq z}

In the following exercises, find the volume of the solid E whose boundaries are given in rectangular coordinates. \(E=\left\{(x, y, z) \mid \sqrt{x^{2}+y^{2}} \leq z \leq \sqrt{16-x^{2}-y^{2}}, x \geq 0, y \geq 0\right\}\) Text Transcription: E = {(x, y, z) mid sqrt x^2 + y^2 leq z leq sqrt 16 - x^2 - y^2, x geq 0, y geq 0}

In the following exercises, find the volume of the solid E whose boundaries are given in rectangular coordinates. \(E=\left\{(x, y, z) \mid x^{2}+y^{2}+z^{2}-2 z \leq 0, \sqrt{x^{2}+y^{2}} \leq z\right\}\) Text Transcription: E = {(x, y, z) mid x^2 + y^2 + z^2 - 2z leq 0, sqrt x^2 + y^2 leq z}

Use spherical coordinates to find the volume of the solid situated outside the sphere \(\rho=1\) and inside the sphere \(\rho=\cos \varphi\), with \(\varphi \in\left[0, \frac{\pi}{2}\right]\) . Text Transcription: rho=1 rho=cos varphi varphi in [0, pi/2]

Use spherical coordinates to find the volume of the ball \(\rho \leq 3\) that is situated between the cones \(\varphi=\frac{\pi}{4}\) and \(\varphi=\frac{\pi}{3}\) . Text Transcription: rho leq 3 varphi=pi/4 varphi=pi/3

Convert the \(\int_{-4}^{4} \int_{-\sqrt{16-y^{2}}}^{\sqrt{16-y^{2}}} \int_{-\sqrt{16-x^{2}-y^{2}}}^{\sqrt{16-x^{2}-y^{2}}} \left(x^{2}+y^{2}+z^{2}\right) d z d x d y\) into an integral in spherical coordinates. Text Transcription: int_-4^4 int_-sqrt 16-y^2 ^ sqrt 16-y^2 int_-sqrt 16-x^2-y^2 ^ sqrt 16-x^2-y^2 (x^2+y^2+z^2) dzdxdy

Convert the \(\int_{0}^{4} \int_{0}^{\sqrt{16-x^{2}}} \int_{-\sqrt{16-x^{2}-y^{2}}}^{\sqrt{16-x^{2}}}\left(x^{2}+y^{2}+z^{2}\right)^{2} d z d y d x) into an integral in spherical coordinates. Text Transcription: int_0^4 int_0 ^ sqrt 16-x^2 int_-sqrt 16-x^2-y^2 ^ sqrt 16-x^2 (x^2+y^2+z^2)^2 dzdydx

Convert the integral \(\int_{-2}^{2} \int_{-\sqrt{4-x^{2}}}^{\sqrt{4-x^{2}}} \int_{ \sqrt{x^{2}+y^{2}}}^{\sqrt{16-x^{2}-y^{2}}} d z d y d x\) into an integral in spherical coordinates and evaluate it. Text Transcription: int_-2 ^ 2 int_-sqrt 4-x^2 ^ sqrt 4-x^2 int_sqrt x^2+y^2 ^ sqrt 16-x^2-y^2 dzdydx

[T] Use a CAS to graph the solid whose volume is given by the iterated integral in spherical coordinates \(\int_{\pi / 2}^{\pi} \int_{5 \pi / 6}^{\pi / 6} \int_{0}^{2} \rho^{2} \sin \varphi d \rho d \varphi d \theta\). Find the volume V of the solid. Round your answer to three decimal places. Text Transcription: int_pi/2 ^ pi int_5 pi/6 ^ pi / 6 int_0 ^ 2 rho^2 sin varphi d rho d varphi d theta

[T] Use a CAS to graph the solid whose volume is given by the iterated integral in spherical coordinates as \(\int_{0}^{2 \pi} \int_{3 \pi / 4}^{\pi / 4} \int_{0}^{1} \rho^{2} \sin \varphi d \rho d \varphi d \theta\). Find the volume V of the solid. Round your answer to three decimal places. Text Transcription: int_0 ^ 2 pi int_3 pi/4 ^ pi/4 int_0 ^ 1 rho^2 sin varphi d rho d varphi d theta

[T] Use a CAS to evaluate the integral \(\iiint_{E}\left(x^{2}+y^{2}\right) d V\) where E lies above the paraboloid \(z=x^{2}+y^{2}\) and below the plane z = 3y. Text Transcription: iiint_E (x^2+y^2) dV z=x^2+y^2

[T] a. Evaluate the integral \(\iiint_{E} e^{\sqrt{x^{2}+y^{2}+z^{2}}} d V\), where E is bounded by the spheres \(4 x^{2}+4 y^{2}+4 z^{2}=1\) and \(x^{2}+y^{2}+z^{2}=1\). b. Use a CAS to find an approximation of the previous integral. Round your answer to two decimal places. Text Transcription: iiint_E e^sqrt x^2+y^2+z^2 dV 4x^2+4y^2+4z^2=1 x^2+y^2+z^2=1

Express the volume of the solid inside the sphere \(x^{2}+y^{2}+z^{2}=16\) and outside the cylinder \(x^{2}+y^{2}=4\) as triple integrals in cylindrical coordinates and spherical coordinates, respectively. Text Transcription: x^2+y^2+z^2=16 x^2+y^2=4

Express the volume of the solid inside the sphere \(x^{2}+y^{2}+z^{2}=16\) and outside the cylinder \(x^{2}+y^{2}=4\) that is located in the first octant as triple integrals in cylindrical coordinates and spherical coordinates, respectively. Text Transcription: x^2+y^2+z^2=16 x^2+y^2=4

The power emitted by an antenna has a power density per unit volume given in spherical coordinates by \(p(\rho, \theta, \varphi)=\frac{P_{0}}{\rho^{2}} \cos^{2} \theta \sin^{4} \varphi\), where \(P_{0}\) is a constant with units in watts. The total power within a sphere B of radius r meters is defined as \(P=\iiint_{B} p(\rho, \theta, \varphi) d V\). Find the total power P. Text Transcription: p(rho, theta, varphi)=P_0 / rho^2 cos^2 theta sin^4 varphi P_0 P=iiint_B p(rho, theta, varphi) dV

Use the preceding exercise to find the total power within a sphere B of radius 5 meters when the power density per unit volume is given by \(p(\rho, \theta, \varphi)=\frac{30}{\rho^{2}} \cos ^{2} \theta \sin ^{4} \varphi\) . Text Transcription: p(rho, theta, varphi)=frac30 / rho^2 cos^2 theta sin^4 varphi

A charge cloud contained in a sphere B of radius r centimeters centered at the origin has its charge density given by \(q(x, y, z)=k \sqrt{x^{2}+y^{2}+z^{2}} \frac{\mu \mathrm{C}}{\mathrm{cm}^{3}}\) , where k > 0 . The total charge contained in B is given by \(Q=\iiint_{R} q(x, y, z) d V\) . Find the total charge Q . Text Transcription: q(x, y, z)=k sqrt x^2+y^2+z^2 mu C / cm^3 Q=iiint_R q(x, y, z) dV

Use the preceding exercise to find the total charge cloud contained in the unit sphere if the charge density is \(q(x, y, z)=20 \sqrt{x^{2}+y^{2}+z^{2}} \frac{\mu \mathrm{C}}{\mathrm{cm}^{3}} .\) . Text Transcription: q(x, y, z)=20 sqrt x^2+y^2+z^2 mu C / cm^3

In the following exercises, find the volume of the solid E whose boundaries are given in rectangular coordinates.

a. Let B be the region between the upper concentric hemispheres of radii a and b centered at the origin and situated in the first octant, where 0 < a < b. Consider F a function defined on B whose form in spherical coordinates \((\rho, \theta, \varphi)\) is \(F(x, y, z)=f(\rho) \cos \varphi\). Show that if g(a) = g(b) = 0 and \(\int_{a}^{b} h(\rho) d \rho=0\), then \(\iiint_{B} F(x, y, z) d V=\frac{\pi^{2}}{4}[a h(a)-b h(b)]\), where g is an antiderivative of f and h is an antiderivative of g. b. Use the previous result to show that \(\iiint_{B} \frac{z \cos \sqrt{x^{2}+y^{2}+z^{2}}}{\sqrt{x^{2}+y^{2}+z^{2}}} d V=\frac{3 \pi^{2}}{2}\) , where B is the region between the upper concentric hemispheres of radii \(\pi\) and \(2 \pi\) centered at the origin and situated in the first octant. Text Transcription: (rho, theta, varphi) F(x, y, z) = f(rho) cos varphi int_a^b h(rho) d rho = 0 iiint_B F(x, y, z) dV = pi^2 / 4 [a h(a)-b h(b)] iiint_B z cos sqrt x^2 + y^2 + z^2 / sqrt x^2 + y^2 + z^2 dV =3 pi^2 / 2 pi 2pi

In the following exercises, the function f and region E are given. a. Express the region E and function f in cylindrical coordinates. b. Convert the integral \(\iiint_{B} f(x, y, z) d V\) into cylindrical coordinates and evaluate it. f(x, y, z) = z; \(E=\left\{(x, y, z) \mid 0 \leq x^{2}+y^{2}+z^{2} \leq 1, z \geq 0\right\}\) Text Transcription: iiint_B f(x, y, z) dV E = {(x, y, z) mid 0 leq x^2 + y^2 + z^2 leq 1, z geq 0}

In the following exercises, the function f and region E are given. a. Express the region E and function f in cylindrical coordinates. b. Convert the integral \(\iiint_{B} f(x, y, z) d V\) into cylindrical coordinates and evaluate it. f(x, y, z) = x + y; \(E=\left\{(x, y, z) \mid 1 \leq x^{2}+y^{2}+z^{2} \leq 2, z \geq 0, y \geq 0\right\}\) Text Transcription: iiint_B f(x, y, z) dV E = {(x, y, z) mid 1 leq x^2 + y^2 + z^2 leq 2, z geq 0, y geq 0}

In the following exercises, the function f and region E are given. a. Express the region E and function f in cylindrical coordinates. b. Convert the integral \(\iiint_{B} f(x, y, z) d V\) into cylindrical coordinates and evaluate it. f(x, y, z) = 2xy; \(E=\left\{(x, y, z) \mid \sqrt{x^{2}+y^{2}} \leq z \leq \sqrt{1-x^{2}-y^{2}}, x \geq 0, y \geq 0\right\}\) Text Transcription: iiint_B f(x, y, z) dV E = {(x, y, z) mid sqrt x^2 + y^2 leq z leq sqrt 1 - x^2 - y^2, x geq 0, y geq 0}

In the following exercises, the function f and region E are given. a. Express the region E and function f in cylindrical coordinates. b. Convert the integral \(\iiint_{B} f(x, y, z) d V\) into cylindrical coordinates and evaluate it. f(x, y, z) = z; \(E=\left\{(x, y, z) \mid x^{2}+y^{2}+z^{2}-2 z \leq 0, \sqrt{x^{2}+y^{2}} \leq z\right\}\) Text Transcription: iiint_B f(x, y, z) dV E = {(x, y, z) mid x^2 + y^2 + z^2 - 2z leq 0, sqrt x^2 + y^2 leq z}

In the following exercises, find the volume of the solid E whose boundaries are given in rectangular coordinates. \(E=\left\{(x, y, z) \mid \sqrt{x^{2}+y^{2}} \leq z \leq \sqrt{16-x^{2}-y^{2}}, x \geq 0, y \geq 0\right\}\) Text Transcription: E = {(x, y, z) mid sqrt x^2 + y^2 leq z leq sqrt 16 - x^2 - y^2, x geq 0, y geq 0}

In the following exercises, find the volume of the solid E whose boundaries are given in rectangular coordinates. \(E=\left\{(x, y, z) \mid x^{2}+y^{2}+z^{2}-2 z \leq 0, \sqrt{x^{2}+y^{2}} \leq z\right\}\) Text Transcription: E = {(x, y, z) mid x^2 + y^2 + z^2 - 2z leq 0, sqrt x^2 + y^2 leq z}

Use spherical coordinates to find the volume of the solid situated outside the sphere \(\rho=1\) and inside the sphere \(\rho=\cos \varphi\), with \(\varphi \in\left[0, \frac{\pi}{2}\right]\) . Text Transcription: rho=1 rho=cos varphi varphi in [0, pi/2]

Use spherical coordinates to find the volume of the ball \(\rho \leq 3\) that is situated between the cones \(\varphi=\frac{\pi}{4}\) and \(\varphi=\frac{\pi}{3}\) . Text Transcription: rho leq 3 varphi=pi/4 varphi=pi/3

Convert the \(\int_{-4}^{4} \int_{-\sqrt{16-y^{2}}}^{\sqrt{16-y^{2}}} \int_{-\sqrt{16-x^{2}-y^{2}}}^{\sqrt{16-x^{2}-y^{2}}} \left(x^{2}+y^{2}+z^{2}\right) d z d x d y\) into an integral in spherical coordinates. Text Transcription: int_-4^4 int_-sqrt 16-y^2 ^ sqrt 16-y^2 int_-sqrt 16-x^2-y^2 ^ sqrt 16-x^2-y^2 (x^2+y^2+z^2) dzdxdy

Convert the \(\int_{0}^{4} \int_{0}^{\sqrt{16-x^{2}}} \int_{-\sqrt{16-x^{2}-y^{2}}}^{\sqrt{16-x^{2}}}\left(x^{2}+y^{2}+z^{2}\right)^{2} d z \ d y \ d x)\) into an integral in spherical coordinates. Text Transcription: int_0^4 int_0 ^ sqrt 16-x^2 int_-sqrt 16-x^2-y^2 ^ sqrt 16-x^2 (x^2+y^2+z^2)^2 dzdydx

Convert the integral \(\int_{-2}^{2} \int_{-\sqrt{4-x^{2}}}^{\sqrt{4-x^{2}}} \int_{ \sqrt{x^{2}+y^{2}}}^{\sqrt{16-x^{2}-y^{2}}} d z d y d x\) into an integral in spherical coordinates and evaluate it. Text Transcription: int_-2 ^ 2 int_-sqrt 4-x^2 ^ sqrt 4-x^2 int_sqrt x^2+y^2 ^ sqrt 16-x^2-y^2 dzdydx

[T] Use a CAS to graph the solid whose volume is given by the iterated integral in spherical coordinates \(\int_{\pi / 2}^{\pi} \int_{5 \pi / 6}^{\pi / 6} \int_{0}^{2} \rho^{2} \sin \varphi d \rho d \varphi d \theta\). Find the volume V of the solid. Round your answer to three decimal places. Text Transcription: int_pi/2 ^ pi int_5 pi/6 ^ pi / 6 int_0 ^ 2 rho^2 sin varphi d rho d varphi d theta

[T] Use a CAS to graph the solid whose volume is given by the iterated integral in spherical coordinates as \(\int_{0}^{2 \pi} \int_{3 \pi / 4}^{\pi / 4} \int_{0}^{1} \rho^{2} \sin \varphi \ d \rho \ d \varphi \ d \theta\). Find the volume V of the solid. Round your answer to three decimal places. Text Transcription: int_0 ^ 2 pi int_3 pi/4 ^ pi/4 int_0 ^ 1 rho^2 sin varphi d rho d varphi d theta

[T] Use a CAS to evaluate the integral \(\iiint_{E}\left(x^{2}+y^{2}\right) d V\) where E lies above the paraboloid \(z=x^{2}+y^{2}\) and below the plane z = 3y. Text Transcription: iiint_E (x^2+y^2) dV z=x^2+y^2

[T] a. Evaluate the integral \(\iiint_{E} e^{\sqrt{x^{2}+y^{2}+z^{2}}} d V\), where E is bounded by the spheres \(4 x^{2}+4 y^{2}+4 z^{2}=1\) and \(x^{2}+y^{2}+z^{2}=1\). b. Use a CAS to find an approximation of the previous integral. Round your answer to two decimal places. Text Transcription: iiint_E e^sqrt x^2+y^2+z^2 dV 4x^2+4y^2+4z^2=1 x^2+y^2+z^2=1

Express the volume of the solid inside the sphere \(x^{2}+y^{2}+z^{2}=16\) and outside the cylinder \(x^{2}+y^{2}=4\) as triple integrals in cylindrical coordinates and spherical coordinates, respectively. Text Transcription: x^2+y^2+z^2=16 x^2+y^2=4

Express the volume of the solid inside the sphere \(x^{2}+y^{2}+z^{2}=16\) and outside the cylinder \(x^{2}+y^{2}=4\) that is located in the first octant as triple integrals in cylindrical coordinates and spherical coordinates, respectively. Text Transcription: x^2+y^2+z^2=16 x^2+y^2=4

The power emitted by an antenna has a power density per unit volume given in spherical coordinates by \(p(\rho, \theta, \varphi)=\frac{P_{0}}{\rho^{2}} \cos^{2} \theta \sin^{4} \varphi\), where \(P_{0}\) is a constant with units in watts. The total power within a sphere B of radius r meters is defined as \(P=\iiint_{B} p(\rho, \theta, \varphi) d V\). Find the total power P. Text Transcription: p(rho, theta, varphi)=P_0 / rho^2 cos^2 theta sin^4 varphi P_0 P=iiint_B p(rho, theta, varphi) dV

Use the preceding exercise to find the total power within a sphere B of radius 5 meters when the power density per unit volume is given by \(p(\rho, \theta, \varphi)=\frac{30}{\rho^{2}} \cos ^{2} \theta \sin ^{4} \varphi\) . Text Transcription: p(rho, theta, varphi)=frac30 / rho^2 cos^2 theta sin^4 varphi

A charge cloud contained in a sphere B of radius r centimeters centered at the origin has its charge density given by \(q(x, y, z)=k \sqrt{x^{2}+y^{2}+z^{2}} \frac{\mu \mathrm{C}}{\mathrm{cm}^{3}}\) , where k > 0 . The total charge contained in B is given by \(Q=\iiint_{R} q(x, y, z) d V\) . Find the total charge Q . Text Transcription: q(x, y, z)=k sqrt x^2+y^2+z^2 mu C / cm^3 Q=iiint_R q(x, y, z) dV

Use the preceding exercise to find the total charge cloud contained in the unit sphere if the charge density is \(q(x, y, z)=20 \sqrt{x^{2}+y^{2}+z^{2}} \frac{\mu \mathrm{C}}{\mathrm{cm}^{3}} \) . Text Transcription: q(x, y, z)=20 sqrt x^2+y^2+z^2 mu C / cm^3