Volume In Exercises 3336, use spherical coordinates to findthe volume of the solid

In Exercises 1–6, evaluate the iterated integral. \(\int_{-1}^{5} \int_{0}^{\pi / 2} \int_{0}^{3} r \cos \theta d r d \theta d z\) Text Transcription: int_-1 ^5 int_0 ^pi / 2 int_0 ^3 r cos theta dr dtheta dz

In Exercises 1–6, evaluate the iterated integral. \(\int_{0}^{\pi / 4} \int_{0}^{6} \int_{0}^{6-r} r z d z d r d \theta\) Text Transcription: int_0 ^pi / 4 int_0 ^6 int_0 ^6 - r rz dz dr dtheta

In Exercises 1–6, evaluate the iterated integral. \(\int_{0}^{\pi / 2} \int_{0}^{2 \cos ^{2} \theta} \int_{0}^{4-r^{2}} r \sin \theta d z d r d \theta\) Text Transcription: int_0 ^pi / 2 int_0 ^2 cos^2 theta int_0 ^4 - r^2 r sin theta dz dr dtheta

In Exercises 1–6, evaluate the iterated integral. \(\int_{0}^{\pi / 2} \int_{0}^{\pi} \int_{0}^{2} e^{-\rho^{3}} \rho^{2} d \rho d \theta d \phi\) Text Transcription: int_0 ^pi / 2 int_0 ^pi int_0 ^2 e^- rho^3 rho^2 d rho d theta d phi

In Exercises 1–6, evaluate the iterated integral. \(\int_{0}^{2 \pi} \int_{0}^{\pi / 4} \int_{0}^{\cos \phi} \rho^{2} \sin \phi d \rho d \phi d \theta\) Text Transcription: int_0 ^2 \pi int_0 ^pi / 4 int_0 ^cos phi rho^2 sin phi d rho d phi d theta

In Exercises 1–6, evaluate the iterated integral. \(\int_{0}^{\pi / 4} \int_{0}^{\pi / 4} \int_{0}^{\cos \theta} \rho^{2} \sin \phi \cos \phi d \rho d \theta d \phi\) Text Transcription: int_0 ^pi / 4 int_0 ^pi / 4 int_0 ^cos theta rho^2 sin phi cos phi d rho d theta d phi

In Exercises 7 and 8, use a computer algebra system to evaluate the iterated integral. \(\int_{0}^{4} \int_{0}^{z} \int_{0}^{\pi / 2} r e^{r} d \theta d r d z\) Text Transcription: int_0 ^4 int_0 ^z int_0 ^pi / 2 re^r d theta dr dz

In Exercises 7 and 8, use a computer algebra system to evaluate the iterated integral. \(\int_{0}^{\pi / 2} \int_{0}^{\pi} \int_{0}^{\sin \theta}(2 \cos \phi) \rho^{2} d \rho d \theta d \phi\) Text Transcription: int_0 ^pi / 2 int_0 ^pi int_0 ^sin theta (2 cos phi) rho^2 d rho d theta d phi

In Exercises 9–12, sketch the solid region whose volume is given by the iterated integral, and evaluate the iterated integral. \(\int_{0}^{\pi / 2} \int_{0}^{3} \int_{0}^{e^{-r^{2}}} r d z d r d \theta\) Text Transcription: int_0 ^pi / 2 int_0 ^3 int_0 ^e^-r^2 r dz dr d theta

In Exercises 9–12, sketch the solid region whose volume is given by the iterated integral, and evaluate the iterated integral. \(\int_{0}^{2 \pi} \int_{0}^{\sqrt{5}} \int_{0}^{5-r^{2}} r d z d r d \theta\) Text Transcription: int_0 ^2 pi int_0 ^sqrt 5 int_0 ^5 - r^2 r dz dr d theta

In Exercises 9–12, sketch the solid region whose volume is given by the iterated integral, and evaluate the iterated integral. \(\int_{0}^{2 \pi} \int_{\pi / 6}^{\pi / 2} \int_{0}^{4} \rho^{2} \sin \phi d \rho d \phi d \theta\) Text Transcription: int_0 ^2 pi int_pi / 6 ^pi / 2 int_0 ^4 rho^2 sin phi d rho d phi d theta

In Exercises 9–12, sketch the solid region whose volume is given by the iterated integral, and evaluate the iterated integral. \(\int_{0}^{2 \pi} \int_{0}^{\pi} \int_{2}^{5} \rho^{2} \sin \phi d \rho d \phi d \theta\) Text Transcription: int_0 ^2 pi int_0 ^pi int_2 ^5 rho^2 sin phi d rho d phi d theta

In Exercises 13–16, convert the integral from rectangular coordinates to both cylindrical and spherical coordinates, and evaluate the simplest iterated integral. \(\int_{-2}^{2} \int_{-\sqrt{4-x^{2}}}^{\sqrt{4-x^{2}}} \int_{x^{2}+y^{2}}^{4} x d z d y d x\) Text Transcription: int_-2 ^2 int_-sqrt 4 - x^2 ^sqrt 4 - x^2 int_x ^2 + y^2 ^4 x dz dy dx

In Exercises 13–16, convert the integral from rectangular coordinates to both cylindrical and spherical coordinates, and evaluate the simplest iterated integral. \(\int_{0}^{2} \int_{0}^{\sqrt{4-x^{2}}} \int_{0}^{\sqrt{16-x^{2}-y^{2}}} \sqrt{x^{2}+y^{2}} d z d y d x\) Text Transcription: int_0 ^2 int_0 ^sqrt 4 - x^2 int_0 ^sqrt 16 - x^2 - y^2 sqrt x^2 + y^2 dz dy dx

In Exercises 13–16, convert the integral from rectangular coordinates to both cylindrical and spherical coordinates, and evaluate the simplest iterated integral. \(\int_{-a}^{a} \int_{-\sqrt{a^{2}-x^{2}}}^{\sqrt{a^{2}-x^{2}}} \int_{a}^{a+\sqrt{a^{2}-x^{2}-y^{2}}} x d z d y d x\) Text Transcription: int_-a ^a int_-sqrt a^2 - x^2 ^sqrt a^2 - x^2 int_a ^a + sqrt a^2 - x^2 - y^2 x dz dy dx

In Exercises 13–16, convert the integral from rectangular coordinates to both cylindrical and spherical coordinates, and evaluate the simplest iterated integral. \(\int_{0}^{3} \int_{0}^{\sqrt{9-x^{2}}} \int_{0}^{\sqrt{9-x^{2}-y^{2}}} \sqrt{x^{2}+y^{2}+z^{2}} d z d y d x\) Text Transcription: int_0 ^3 int_0 ^sqrt 9 - x^2 int_0 ^sqrt 9 - x^2 - y^2 sqrt x^2 + y^2 + z^2 dz dy dx

Volume In Exercises 17–22, use cylindrical coordinates to find the volume of the solid. Solid inside both \(x^{2}+y^{2}+z^{2}=a^{2}\) and \((x-a / 2)^{2}+y^{2}=(a / 2)^{2}\) Text Transcription: x^2 + y^2 + z^2 = a^2 (x - a / 2)^2 + y^2 = (a / 2)^2

Volume In Exercises 17–22, use cylindrical coordinates to find the volume of the solid. Solid inside \(x^{2}+y^{2}+z^{2}=16\) and outside \(z=\sqrt{x^{2}+y^{2}}\) Text Transcription: x^2 + y^2 + z^2 = 16 z = sqrt x^2 + y^2

Volume In Exercises 17–22, use cylindrical coordinates to find the volume of the solid. Solid bounded above by z = 2x and below by \(z=2 x^{2}+2 y^{2}\) Text Transcription: z = 2x^2 + 2y^2

Volume In Exercises 17–22, use cylindrical coordinates to find the volume of the solid. Solid bounded above by \(z=2-x^{2}-y^{2}\) and below by \(z=x^{2}+y^{2}\) Text Transcription: z = 2 - x^2 - y^2 z = x^2 + y^2

Volume In Exercises 17–22, use cylindrical coordinates to find the volume of the solid. Solid bounded by the graphs of the sphere \(r^{2}+z^{2}=a^{2}\) and the cylinder \(r=a \cos \theta\) Text Transcription: r^{2}+z^{2}=a^{2} r = a cos theta

Volume In Exercises 17–22, use cylindrical coordinates to find the volume of the solid. Solid inside the sphere \(x^{2}+y^{2}+z^{2}=4\) and above the upper nappe of the cone \(z^{2}=x^{2}+y^{2}\) Text Transcription: x^2 + y^2 + z^2 = 4 z^2 = x^2 + y^2

Mass In Exercises 23 and 24, use cylindrical coordinates to find the mass of the solid Q. \(Q=\left\{(x, y, z): 0 \leq z \leq 9-x-2 y, x^{2}+y^{2} \leq 4\right\}\) \(\rho(x, y, z)=k \sqrt{x^{2}+y^{2}}\) Text Transcription: Q=\left\{(x, y, z): 0 \leq z \leq 9-x-2 y, x^{2}+y^{2} \leq 4\right\} \rho(x, y, z)=k \sqrt{x^{2}+y^{2}}

Mass In Exercises 23 and 24, use cylindrical coordinates to find the mass of the solid Q. \(Q=\left\{(x, y, z): 0 \leq z \leq 12 e^{-\left(x^{2}+y^{2}\right)}, x^{2}+y^{2} \leq 4, x \geq 0, y \geq 0\right\}\) \(\rho(x, y, z)=k\) Text Transcription: Q=\left\{(x, y, z): 0 \leq z \leq 12 e^{-\left(x^{2}+y^{2}\right)}, x^{2}+y^{2} \leq 4, x \geq 0, y \geq 0\right\} \rho(x, y, z)=k

In Exercises 25–30, use cylindrical coordinates to find the indicated characteristic of the cone shown in the figure. Volume Find the volume of the cone.

In Exercises 25–30, use cylindrical coordinates to find the indicated characteristic of the cone shown in the figure. Centroid Find the centroid of the cone.

In Exercises 25–30, use cylindrical coordinates to find the indicated characteristic of the cone shown in the figure. Center of Mass Find the center of mass of the cone assuming that its density at any point is proportional to the distance between the point and the axis of the cone. Use a computer algebra system to evaluate the triple integral.

In Exercises 25–30, use cylindrical coordinates to find the indicated characteristic of the cone shown in the figure. Center of Mass Find the center of mass of the cone assuming that its density at any point is proportional to the distance between the point and the base. Use a computer algebra system to evaluate the triple integral.

In Exercises 25–30, use cylindrical coordinates to find the indicated characteristic of the cone shown in the figure. Moment of Inertia Assume that the cone has uniform density and show that the moment of inertia about the z-axis is \(I_{z}=\frac{3}{10} m r_{0}^{2}\) Text Transcription: I_z = 3/10 mr_0 ^2

In Exercises 25–30, use cylindrical coordinates to find the indicated characteristic of the cone shown in the figure. Moment of Inertia Assume that the density of the cone is \(\rho(x, y, z)=k \sqrt{x^{2}+y^{2}}\) and find the moment of inertia about the z-axis. Text Transcription: rho(x, y, z)=k sqrt x^2 + y^2

Moment of Inertia In Exercises 31 and 32, use cylindrical coordinates to verify the given formula for the moment of inertia of the solid of uniform density. Cylindrical shell: \(I_{z}=\frac{1}{2} m\left(a^{2}+b^{2}\right)\) \(0<a \leq r \leq b, \quad 0 \leq z \leq h\) Text Transcription: I_z = 1/2 m(a^2 + b^2) 0<a leq r leq b, 0 leq z leq h

Moment of Inertia In Exercises 31 and 32, use cylindrical coordinates to verify the given formula for the moment of inertia of the solid of uniform density. Right circular cylinder: \(I_{z}=\frac{3}{2} m a^{2}\) \(r=2 a \sin \theta, \quad 0 \leq z \leq h\) Use a computer algebra system to evaluate the triple integral. Text Transcription: I_z = 3/2 ma^2 r=2 a sin theta, 0 leq z leq h

Volume In Exercises 33–36, use spherical coordinates to find the volume of the solid. Solid inside \(x^{2}+y^{2}+z^{2}=9\), outside \(z=\sqrt{x^{2}+y^{2}}\), and above the xy-plane Text Transcription: x^2 + y^2 + z^2 = 9 z=sqrt x^2 + y^2

Volume In Exercises 33–36, use spherical coordinates to find the volume of the solid. Solid bounded above by \(x^{2}+y^{2}+z^{2}=z\) and below by \(z=\sqrt{x^{2}+y^{2}}\) Text Transcription: x^2 + y^2 + z^2 = z z = sqrt x^2 + y^2

Volume In Exercises 33–36, use spherical coordinates to find the volume of the solid. The torus given by \(\rho=4 \sin \phi\) (Use a computer algebra system to evaluate the triple integral.) Text Transcription: rho=4 sin phi

Volume In Exercises 33–36, use spherical coordinates to find the volume of the solid. The solid between the spheres \(x^{2}+y^{2}+z^{2}=a^{2}\) and \(x^{2}+y^{2}+z^{2}=b^{2}, b>a\), and inside the cone \(z^{2}=x^{2}+y^{2}\) Text Transcription: x^2 + y^2 + z^2 = a^2 x^2 + y^2 + z^2 = b^2, b>a z^2 = x^2 + y^2

Mass In Exercises 37 and 38, use spherical coordinates to find the mass of the sphere \(x^{2}+y^{2}+z^{2}=a^{2}\) with the given density. The density at any point is proportional to the distance between the point and the origin. Text Transcription: x^2 + y^2 + z^2 = a^2

Mass In Exercises 37 and 38, use spherical coordinates to find the mass of the sphere \(x^{2}+y^{2}+z^{2}=a^{2}\) with the given density. The density at any point is proportional to the distance of the point from the z-axis. Text Transcription: x^2 + y^2 + z^2 = a^2

Center of Mass In Exercises 39 and 40, use spherical coordinates to find the center of mass of the solid of uniform density. Hemispherical solid of radius r

Center of Mass In Exercises 39 and 40, use spherical coordinates to find the center of mass of the solid of uniform density. Solid lying between two concentric hemispheres of radii r and R, where r < R

Moment of Inertia In Exercises 41 and 42, use spherical coordinates to find the moment of inertia about the z-axis of the solid of uniform density. Solid bounded by the hemisphere \(\rho=\cos \phi, \pi / 4 \leq \phi \leq \pi / 2\), and the cone \(\phi=\pi / 4\) Text Transcription: rho=cos phi, pi / 4 leq phi leq pi / 2 phi=pi / 4

Moment of Inertia In Exercises 41 and 42, use spherical coordinates to find the moment of inertia about the z-axis of the solid of uniform density. Solid lying between two concentric hemispheres of radii r and R, where r < R

Give the equations for conversion from rectangular to cylindrical coordinates and vice versa.

Give the equations for conversion from rectangular to spherical coordinates and vice versa.

Give the iterated form of the triple integral \(\iiint_{Q} f(x, y, z) d V\) in cylindrical form. Text Transcription: iiint_Q f(x, y, z) dV

Give the iterated form of the triple integral \(\iiint_{Q} f(x, y, z) d V\) in spherical form. Text Transcription: iiint_Q f(x, y, z) dV

Describe the surface whose equation is a coordinate equal to a constant for each of the coordinates in (a) the cylindrical coordinate system and (b) the spherical coordinate system.

Convert the integral from rectangular coordinates to both (a) cylindrical and (b) spherical coordinates. Without calculating, which integral appears to be the simplest to evaluate? Why? \(\int_{0}^{a} \int_{0}^{\sqrt{a^{2}-x^{2}}} \int_{0}^{\sqrt{a^{2}-x^{2}-y^{2}}} \sqrt{x^{2}+y^{2}+z^{2}} d z d y d x\) Text Transcription: int_0 ^a int_0 ^sqrt a^2 - x^2 int_0 ^sqrt a^2 - x^2 - y^2 sqrt x^2 + y^2 + z^2 dz dy dx

Find the "volume" of the “four-dimensional sphere" \(x^{2}+y^{2}+z^{2}+w^{2}=a^{2}\) by evaluating \(16 \int_{0}^{a} \int_{0}^{\sqrt{a^{2}-x^{2}}} \int_{0}^{\sqrt{a^{2}-x^{2}-y^{2}}} \int_{0}^{\sqrt{a^{2}-x^{2}-y^{2}-z^{2}}} d w d z d y d x\) Text Transcription: x^2 + y^2 + z^2 + w^2 = a^2 16 int_0 ^a int_0 ^sqrt a^2 - x^2 int_0 ^sqrt a^2 - x^2 - y^2 int_0 ^sqrt a^2 - x^2 - y^2 - z^2 dw d z dy dx

Use spherical coordinates to show that \(\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \sqrt{x^{2}+y^{2}+z^{2}} e^{-\left(x^{2}+y^{2}+z^{2}\right)} d x d y d z=2 \pi\) Text Transcription: int_-infty ^infty int_-infty ^infty int_-infty ^infty sqrt x^2 + y^2 + z^2 e^(x^2 + y^2 + z^2) dx dy dz = 2 pi

Find the volume of the region of points (x, y, z) such that \(\left(x^{2}+y^{2}+z^{2}+8\right)^{2} \leq 36\left(x^{2}+y^{2}\right)\). Text Transcription: (x^2 + y^2 + z^2 + 8)^2 leq 36(x^2 + y^2)