Centroid In Exercises 4954, find the centroid of the solidregion bounded by the graphs

In Exercises 1-8, evaluate the iterated integral. \(\int_{0}^{3} \int_{0}^{2} \int_{0}^{1}(x+y+z) d x d z d y\) Text Transcription: int_{0}^{3} int_{0}^{2} int_{0}^{1}(x+y+z) dx dz dy

In Exercises 1-8, evaluate the iterated integral. \(\int_{-1}^{1} \int_{-1}^{1} \int_{-1}^{1} x^{2} y^{2} z^{2} d x d y d z\) Text Transcription: int_{-1}^{1} int_{-1}^{1} int_{-1}^{1} x^{2} y^{2} z^{2} dx dy dz

In Exercises 1-8, evaluate the iterated integral. \(\int_{0}^{1} \int_{0}^{x} \int_{0}^{x y} x d z d y d x\) Text Transcription: int_{0}^{1} int_{0}^{x} int_{0}^{x y} x dz dy dx

In Exercises 1-8, evaluate the iterated integral. \(\int_{0}^{9} \int_{0}^{y / 3} \int_{0}^{\sqrt{y^{2}-9 x^{2}}} z d z d x d y\) Text Transcription: int_{0}^{9} int_{0}^{y / 3} int_{0}^{sqrt{y^{2}-9 x^{2}}} z dz dx dy

In Exercises 1-8, evaluate the iterated integral. \(\int_{1}^{4} \int_{0}^{1} \int_{0}^{x} 2 z e^{-x^{2}} d y d x d z\) Text Transcription: int_{1}^{4} int_{0}^{1} int_{0}^{x} 2ze^{-x^{2}} dy dx dz

In Exercises 1-8, evaluate the iterated integral. \(\int_{1}^{4} \int_{1}^{e^{2}} \int_{0}^{1 / x z} \ln z d y d z d x\) Text Transcription: int_{1}^{4} int_{1}^{e^{2}} int_{0}^{1 / x z} ln z dy dz dx

In Exercises 1-8, evaluate the iterated integral. \(\int_{0}^{4} \int_{0}^{\pi / 2} \int_{0}^{1-x} x \cos y d z d y d x\) Text Transcription: int_{0}^{4} int_{0}^{\pi / 2} int_{0}^{1-x} x cos y dz dy dx

In Exercises 1-8, evaluate the iterated integral. \(\int_{0}^{\pi / 2} \int_{0}^{y / 2} \int_{0}^{1 / y} \sin y d z d x d y\) Text Transcription: int_{0}^{\pi / 2} int_{0}^{y / 2} int_{0}^{1 / y} sin y dz dx dy

In Exercises 9 and 10, use a computer algebra system to evaluate the iterated integral. \(\int_{0}^{3} \int_{-\sqrt{9-y^{2}}}^{\sqrt{9-y^{2}}} \int_{0}^{y^{2}} y d z d x d y\) Text Transcription: int_{0}^{3} int_{-sqrt{9-y^{2}}}^{sqrt{9-y^{2}}} int_{0}^{y^{2}} y dz dx dy

In Exercises 9 and 10, use a computer algebra system to evaluate the iterated integral. \(\int_{0}^{\sqrt{2}} \int_{0}^{\sqrt{2-x^{2}}} \int_{2 x^{2}+y^{2}}^{4-y^{2}} y d z d y d x\) Text Transcription: int_{0}^{sqrt{2}} int_{0}^{sqrt{2-x^{2}}} int_{2 x^{2}+y^{2}}^{4-y^{2}} y dz dy dx

In Exercises 11 and 12, use a computer algebra system to approximate the iterated integral. \(\int_{0}^{2} \int_{0}^{\sqrt{4-x^{2}}} \int_{1}^{4} \frac{x^{2} \sin y}{z} d z d y d x\) Text Transcription: int_{0}^{2} int_{0}^{sqrt{4-x^{2}}} int_{1}^{4} {x^{2} sin y / z} dz dy dx

In Exercises 11 and 12, use a computer algebra system to approximate the iterated integral. \(\int_{0}^{3} \int_{0}^{2-(2 y / 3)} \int_{0}^{6-2 y-3 z} z e^{-x^{2} y^{2}} d x d z d y\) Text Transcription: int_{0}^{3} int_{0}^{2-(2 y / 3)} int_{0}^{6-2 y-3 z} ze^{-x^{2} y^{2}} dx dz dy

In Exercises 13-18, set up a triple integral for the volume of the solid. The solid in the first octant bounded by the coordinate planes and the plane z = 5 - x - y

In Exercises 13-18, set up a triple integral for the volume of the solid. The solid bounded by \(z=9-x^{2}\), z = 0, y = 0, and y = 2x Text Transcription: z=9-x^{2}

In Exercises 13-18, set up a triple integral for the volume of the solid. The solid bounded by \(z=6-x^{2}-y^{2}\) and z = 0 Text Transcription: z=6-x^{2}-y^{2}

In Exercises 13-18, set up a triple integral for the volume of the solid. The solid bounded by \(z=\sqrt{16-x^{2}-y^{2}}\) and z = 0 Text Transcription: z=sqrt{16-x^{2}-y^{2}}

In Exercises 13-18, set up a triple integral for the volume of the solid. The solid that is the common interior below the sphere \(x^{2}+y^{2}+z^{2}=80\) and above the paraboloid (z=\frac{1}{2}\left(x^{2}+y^{2}\right)\) Text Transcription: x^{2}+y^{2}+z^{2}=80 z=1/2 (x^{2}+y^{2})

In Exercises 13-18, set up a triple integral for the volume of the solid. The solid bounded above by the cylinder \(z=4-x^{2}\) and below by the paraboloid \(z=x^{2}+3 y^{2}\) Text Transcription: z=4-x^{2} z=x^{2}+3 y^{2}

Volume In Exercises 19-22, use a triple integral to find the volume of the solid shown in the figure.

Volume In Exercises 19-22, use a triple integral to find the volume of the solid shown in the figure.

Volume In Exercises 19-22, use a triple integral to find the volume of the solid shown in the figure.

Volume In Exercises 19-22, use a triple integral to find the volume of the solid shown in the figure.

Volume In Exercises 23-26, use a triple integral to find the volume of the solid bounded by the graphs of the equations. \(z=4-x^{2}, y=4-x^{2}\), first octant Text Transcription: z=4-x^{2}, y=4-x^{2}

Volume In Exercises 23-26, use a triple integral to find the volume of the solid bounded by the graphs of the equations. \(z=9-x^{3}, y=-x^{2}+2, y=0, z=0, x \geq 0\) Text Transcription: z=9-x^{3}, y=-x^{2}+2, y=0, z=0, x geq 0

Volume In Exercises 23-26, use a triple integral to find the volume of the solid bounded by the graphs of the equations. \(z=2-y, z=4-y^{2}, x=0, x=3, y=0\) Text Transcription: z=2-y, z=4-y^{2}, x=0, x=3, y=0

Volume In Exercises 23-26, use a triple integral to find the volume of the solid bounded by the graphs of the equations. \(z=x, y=x+2, y=x^{2}\), first octant Text Transcription: z=x, y=x+2, y=x^{2}

In Exercises 27–32, sketch the solid whose volume is given by the iterated integral and rewrite the integral using the indicated order of integration. \(\int_{0}^{1} \int_{-1}^{0} \int_{0}^{y^{2}} d z d y d x\) Rewrite using the order dy dz dx. Text Transcription: int_{0}^{1} int_{-1}^{0} int_{0}^{y^{2}} dz dy dx

In Exercises 27–32, sketch the solid whose volume is given by the iterated integral and rewrite the integral using the indicated order of integration. \(\int_{-1}^{1} \int_{y^{2}}^{1} \int_{0}^{1-x} d z d x d y\) Rewrite using the order dx dz dy. Text Transcription: int_{-1}^{1} int_{y^{2}}^{1} int_{0}^{1-x} dz dx dy

In Exercises 27–32, sketch the solid whose volume is given by the iterated integral and rewrite the integral using the indicated order of integration. \(\int_{0}^{4} \int_{0}^{(4-x) / 2} \int_{0}^{(12-3 x-6 y) / 4} d z d y d x\) Rewrite using the order dy dx dz. Text Transcription: int_{0}^{4} int_{0}^{(4-x) / 2} int_{0}^{(12-3 x-6 y) / 4} dz dy dx

In Exercises 27–32, sketch the solid whose volume is given by the iterated integral and rewrite the integral using the indicated order of integration. \(\int_{0}^{3} \int_{0}^{\sqrt{9-x^{2}}} \int_{0}^{6-x-y} d z d y d x\) Rewrite using the order dz dx dy. Text Transcription: int_{0}^{3} int_{0}^{sqrt{9-x^{2}}} int_{0}^{6-x-y} dz dy dx

In Exercises 27–32, sketch the solid whose volume is given by the iterated integral and rewrite the integral using the indicated order of integration. \(\int_{0}^{1} \int_{y}^{1} \int_{0}^{\sqrt{1-y^{2}}} d z d x d y\) Rewrite using the order dz dy dx. Text Transcription: int_{0}^{1} int_{y}^{1} int_{0}^{sqrt{1-y^{2}}} dz dx dy

In Exercises 27–32, sketch the solid whose volume is given by the iterated integral and rewrite the integral using the indicated order of integration. \(\int_{0}^{2} \int_{2 x}^{4} \int_{0}^{\sqrt{y^{2}-4 x^{2}}} d z d y d x\) Rewrite using the order dx dy dz. Text Transcription: int_{0}^{2} int_{2 x}^{4} int_{0}^{sqrt{y^{2}-4 x^{2}}} dz dy dx

In Exercises 33–36, list the six possible orders of integration for the triple integral over the solid region Q, \(\iint_{Q} \int x y z d V\). \(Q=\{(x, y, z): 0 \leq x \leq 1,0 \leq y \leq x, 0 \leq z \leq 3\}\) Text Transcription: iint_{Q} int xyz dV Q={(x, y, z): 0 leq x leq 1,0 leq y leq x, 0 leq z leq 3}

In Exercises 33–36, list the six possible orders of integration for the triple integral over the solid region Q, \(\iint_{Q} \int x y z d V\). \(Q=\left\{(x, y, z): 0 \leq x \leq 2, x^{2} \leq y \leq 4,0 \leq z \leq 2-x\right\}\) Text Transcription: iint_{Q} int xyz dV Q={(x, y, z): 0 leq x leq 2, x^{2} leq y leq 4,0 leq z leq 2-x}

In Exercises 33–36, list the six possible orders of integration for the triple integral over the solid region Q, \(\iint_{Q} \int x y z d V\). \(Q=\left\{(x, y, z): x^{2}+y^{2} \leq 9,0 \leq z \leq 4\right\}\) Text Transcription: iint_{Q} int xyz dV Q={(x, y, z): x^{2}+y^{2} leq 9,0 leq z leq 4}

In Exercises 33–36, list the six possible orders of integration for the triple integral over the solid region Q, \(\iint_{Q} \int x y z d V\). \(Q=\left\{(x, y, z): 0 \leq x \leq 1, y \leq 1-x^{2}, 0 \leq z \leq 6\right\}\) Text Transcription: iint_{Q} int xyz dV Q={(x, y, z): 0 leq x leq 1, y leq 1-x^{2}, 0 leq z leq 6}

In Exercises 37 and 38, the figure shows the region of integration for the given integral. Rewrite the integral as an equivalent iterated integral in the five other orders. \(\int_{0}^{1} \int_{0}^{1-y^{2}} \int_{0}^{1-y} d z d x d y\) Text Transcription: int_{0}^{1} int_{0}^{1-y^{2}} int_{0}^{1-y} dz dx dy

In Exercises 37 and 38, the figure shows the region of integration for the given integral. Rewrite the integral as an equivalent iterated integral in the five other orders. \(\int_{0}^{3} \int_{0}^{x} \int_{0}^{9-x^{2}} d z d y d x\) Text Transcription: int_{0}^{3} int_{0}^{x} int_{0}^{9-x^{2}} dz dy dx

Mass and Center of Mass In Exercises 39-42, find the mass and the indicated coordinates of the center of mass of the solid of given density bounded by the graphs of the equations. Find \(\bar{x}\) using \(\rho(x, y, z)=k\). Q: 2x + 3y + 6z = 12, x = 0, y = 0, z = 0 Text Transcription: bar{x} rho(x, y, z)=k

Mass and Center of Mass In Exercises 39-42, find the mass and the indicated coordinates of the center of mass of the solid of given density bounded by the graphs of the equations. Find \(\bar{y}\) using \(\rho(x, y, z)=k y\). Q: 3x + 3y + 5z = 15, x = 0, y = 0, z = 0 Text Transcription: bar{y} rho(x, y, z)=k y

Mass and Center of Mass In Exercises 39-42, find the mass and the indicated coordinates of the center of mass of the solid of given density bounded by the graphs of the equations. Find \(\bar{z}\) using \(\rho(x, y, z)=k x\). Q: 7 = 4 – x, z = 0, y = 0, y = 4, x = 0 Text Transcription: bar{z} rho(x, y, z)=kx

Mass and Center of Mass In Exercises 39-42, find the mass and the indicated coordinates of the center of mass of the solid of given density bounded by the graphs of the equations. Find \(\bar{y}\) using \(\rho(x, y, z)=k\). \(Q: \frac{x}{a}+\frac{y}{b}+\frac{z}{c}=1(a, b, c>0), x=0, y=0, z=0\) Text Transcription: bar{y} rho(x, y, z)=k Q: x/a + y/b + z/c = 1(a, b, c > 0), x=0, y=0, z=0

Mass and Center of Mass In Exercises 43 and 44, set up the triple integrals for finding the mass and the center of mass of the solid bounded by the graphs of the equations. x = 0, x = b, y = 0, y = b, z = 0, z = b \(\rho(x, y, z)=k x y\) Text Transcription: rho(x, y, z)=k x y

Mass and Center of Mass In Exercises 43 and 44, set up the triple integrals for finding the mass and the center of mass of the solid bounded by the graphs of the equations. x = 0, x = a, y = 0, y = b, z = 0, z = c \(\rho(x, y, z)=k z\) Text Transcription: rho(x, y, z)=k z

Think About It The center of mass of a solid of constant density is shown in the figure. In Exercises 45-48, make a conjecture about how the center of mass \((\bar{x}, \bar{y}, \bar{z})\) will change for the nonconstant density \(\rho(x, y, z)\). Explain. \(\rho(x, y, z)=k x\) Text Transcription: (bar{x}, bar{y}, bar{z}) rho(x, y, z) rho(x, y, z)=k x

Think About It The center of mass of a solid of constant density is shown in the figure. In Exercises 45-48, make a conjecture about how the center of mass \((\bar{x}, \bar{y}, \bar{z})\) will change for the nonconstant density \(\rho(x, y, z)\). Explain. \(\rho(x, y, z)=k z\) Text Transcription: (bar{x}, bar{y}, bar{z}) rho(x, y, z) rho(x, y, z)=kz

Think About It The center of mass of a solid of constant density is shown in the figure. In Exercises 45-48, make a conjecture about how the center of mass \((\bar{x}, \bar{y}, \bar{z})\) will change for the nonconstant density \(\rho(x, y, z)\). Explain. \(\rho(x, y, z)=k(y+2)\) Text Transcription: (bar{x}, bar{y}, bar{z}) rho(x, y, z) rho(x, y, z)=k(y+2)

Think About It The center of mass of a solid of constant density is shown in the figure. In Exercises 45-48, make a conjecture about how the center of mass \((\bar{x}, \bar{y}, \bar{z})\) will change for the nonconstant density \(\rho(x, y, z)\). Explain. \(\rho(x, y, z)=k x z^{2}(y+2)^{2}\) Text Transcription: (bar{x}, bar{y}, bar{z}) rho(x, y, z) rho(x, y, z)=k x z^{2}(y+2)^{2}

Centroid In Exercises 49-54, find the centroid of the solid region bounded by the graphs of the equations or described by the figure. Use a computer algebra system to evaluate the triple integrals. (Assume uniform density and find the center of mass.) \(z=\frac{h}{r} \sqrt{x^{2}+y^{2}}, z=h\) Text Transcription: z= h/r sqrt{x^{2}+y^{2}}, z=h

Centroid In Exercises 49-54, find the centroid of the solid region bounded by the graphs of the equations or described by the figure. Use a computer algebra system to evaluate the triple integrals. (Assume uniform density and find the center of mass.) \(y=\sqrt{9-x^{2}}, z=y, z=0\) Text Transcription: y=sqrt{9-x^{2}}, z=y, z=0

Centroid In Exercises 49-54, find the centroid of the solid region bounded by the graphs of the equations or described by the figure. Use a computer algebra system to evaluate the triple integrals. (Assume uniform density and find the center of mass.) \(z=\sqrt{16-x^{2}-y^{2}}, z=0\) Text Transcription: z=sqrt{16-x^{2}-y^{2}}, z=0

Centroid In Exercises 49-54, find the centroid of the solid region bounded by the graphs of the equations or described by the figure. Use a computer algebra system to evaluate the triple integrals. (Assume uniform density and find the center of mass.) \(z=\frac{1}{y^{2}+1}, z=0, x=-2, x=2, y=0, y=1\) Text Transcription: z=1/y^{2}+1, z=0, x=-2, x=2, y=0, y=1

Centroid In Exercises 49-54, find the centroid of the solid region bounded by the graphs of the equations or described by the figure. Use a computer algebra system to evaluate the triple integrals. (Assume uniform density and find the center of mass.)

Centroid In Exercises 49-54, find the centroid of the solid region bounded by the graphs of the equations or described by the figure. Use a computer algebra system to evaluate the triple integrals. (Assume uniform density and find the center of mass.)

Moments of Inertia In Exercises 55-58, find \(I_{x}, I_{y}\), and \(I_{z}\), for the solid of given density. Use a computer algebra system to evaluate the triple integrals. (a) \(\rho=k\) (b) \(\rho=k x y z\) Text Transcription: I_{x}, I_{y} I_{z} rho=k rho=k x y z

Moments of Inertia In Exercises 55-58, find \(I_{x}, I_{y}\), and \(I_{z}\), for the solid of given density. Use a computer algebra system to evaluate the triple integrals. (a) \(\rho(x, y, z)=k\) (b) \(\rho(x, y, z)=k\left(x^{2}+y^{2}\right)\) Text Transcription: I_{x}, I_{y} I_{z} rho(x, y, z)=k rho(x, y, z)=k(x^{2}+y^{2})

Moments of Inertia In Exercises 55-58, find \(I_{x}, I_{y}\), and \(I_{z}\), for the solid of given density. Use a computer algebra system to evaluate the triple integrals. (a) \(\rho(x, y, z)=k\) (b) \(\rho=k y\) Text Transcription: I_{x}, I_{y} I_{z} rho(x, y, z)=k rho=ky

Moments of Inertia In Exercises 55-58, find \(I_{x}, I_{y}\), and \(I_{z}\), for the solid of given density. Use a computer algebra system to evaluate the triple integrals. (a) \(\rho=k z\) (b) \(\rho=k(4-z)\) Text Transcription: I_{x}, I_{y} I_{z} rho=kz rho=k(4-z)

Moments of Inertia In Exercises 59 and 60, verify the moments of inertia for the solid of uniform density. Use a computer algebra system to evaluate the triple integrals. \(I_{x}=\frac{1}{12} m\left(3 a^{2}+L^{2}\right)\) \(I_{y}=\frac{1}{2} m a^{2}\) \(I_{z}=\frac{1}{12} m\left(3 a^{2}+L^{2}\right)\) Text Transcription: I_{x}=1/12 m(3 a^{2}+L^{2}) I_{y}=1/2 m a^{2} I_{z}=1/12 m(3 a^{2}+L^{2})

Moments of Inertia In Exercises 59 and 60, verify the moments of inertia for the solid of uniform density. Use a computer algebra system to evaluate the triple integrals. \(I_{x}=\frac{1}{12} m\left(a^{2}+b^{2}\right)\) \(I_{y}=\frac{1}{12} m\left(b^{2}+c^{2}\right)\) \(I_{z}=\frac{1}{12} m\left(a^{2}+c^{2}\right)\) Text Transcription: I_{x}=1/12 m(a^{2}+b^{2}) I_{y}=1/12 m(b^{2}+c^{2}) I_{z}=1/12 m(a^{2}+c^{2})

Moments of Inertia In Exercises 61 and 62, set up a triple integral that gives the moment of inertia about the z-axis of the solid region Q of density \(\rho\). \(Q=\{(x, y, z):-1 \leq x \leq 1,-1 \leq y \leq 1,0 \leq z \leq 1-x\}\) \(\rho=\sqrt{x^{2}+y^{2}+z^{2}}\) Text Transcription: rho Q={(x, y, z): - 1 leq x leq 1, - 1 leq y leq 1,0 leq z leq 1 - x} rho=sqrt x^2 + y^2 + z^2

Moments of Inertia In Exercises 61 and 62, set up a triple integral that gives the moment of inertia about the z-axis of the solid region Q of density \(\rho\). \(Q=\left\{(x, y, z): x^{2}+y^{2} \leq 1,0 \leq z \leq 4-x^{2}-y^{2}\right\}\) \(\rho=k x^{2}\) Text Transcription: rho Q={(x, y, z): x^2 + y^2 leq 1, 0 leq z leq 4 - x^2 - y^2} rho=kx^2

In Exercises 63 and 64, using the description of the solid region, set up the integral for (a) the mass, (b) the center of mass, and (c) the moment of inertia about the z-axis. The solid bounded by \(z=4-x^{2}-y^{2}\) and z = 0 with density function \(\rho=k z\) Text Transcription: z = 4 - x^2 - y^2 rho=k z

In Exercises 63 and 64, using the description of the solid region, set up the integral for (a) the mass, (b) the center of mass, and (c) the moment of inertia about the z-axis. The solid in the first octant bounded by the coordinate planes and \(x^{2}+y^{2}+z^{2}=25\) with density function \(\rho=k x y\) Text Transcription: x^2 + y^2 + z^2 = 25 rho = kxy

Define a triple integral and describe a method of evaluating a triple integral.

Determine whether the moment of inertia about the y-axis of the cylinder in Exercise 59 will increase or decrease for the nonconstant density \(\rho(x, y, z)=\sqrt{x^{2}+z^{2}}\) and a = 4. Text Transcription: rho(x, y, z) = sqrt x^2 + z^2

Consider two solids, solid A and solid B, of equal weight as shown below. (a) Because the solids have the same weight, which has the greater density? (b) Which solid has the greater moment of inertia? Explain. (c) The solids are rolled down an inclined plane. They are started at the same time and at the same height. Which will reach the bottom first? Explain.

Think About It Of the integrals (a)-(c), which one is \(\int_{1}^{3} \int_{0}^{2} \int_{-1}^{1} f(x, y, z) d z d y d x\)? Explain. (a) \(\int_{1}^{3} \int_{0}^{2} \int_{-1}^{1} f(x, y, z) d z d x d y\) (b) \(\int_{-1}^{1} \int_{0}^{2} \int_{1}^{3} f(x, y, z) d x d y d z\) (c) \(\int_{0}^{2} \int_{1}^{3} \int_{-1}^{1} f(x, y, z) d y d x d z\) Text Transcription: int_1 ^3 int_0 ^2 int_-1 ^1 f(x, y, z) dz dy dx int_1 ^3 int_0 ^2 int_-1 ^1 f(x, y, z) dz dx dy int_-1 ^1 int_0 ^2 int_1 ^3 f(x, y, z) dx dy dz int_0 ^2 int_1 ^3 int_-1 ^1 f(x, y, z) dy dx dz

Average Value In Exercises 69-72, find the average value of the function over the given solid. The average value of a continuous function f(x, y, z) over a solid region Q is \(\frac{1}{V} \iint_{Q} \int f(x, y, z) d V\) where V is the volume of the solid region Q. \(f(x, y, z)=z^{2}+4\) over the cube in the first octant bounded by the coordinate planes and the planes x = 1, y = 1, and z = 1 Text Transcription: 1/V iint_Q int f(x, y, z) dV f(x, y, z)=z^2 + 4

Average Value In Exercises 69-72, find the average value of the function over the given solid. The average value of a continuous function f(x, y, z) over a solid region Q is \(\frac{1}{V} \iint_{Q} \int f(x, y, z) d V\) where V is the volume of the solid region Q. f(x, y, z) = xyz over the cube in the first octant bounded by the coordinate planes and the planes x = 4, y = 4, and z = 4 Text Transcription: 1/V iint_Q int f(x, y, z) dV

Average Value In Exercises 69-72, find the average value of the function over the given solid. The average value of a continuous function f(x, y, z) over a solid region Q is \(\frac{1}{V} \iint_{Q} \int f(x, y, z) d V\) where V is the volume of the solid region Q. f(x, y, z) = x + y + z over the tetrahedron in the first octant with vertices (0, 0, 0), (2,0,0), (0, 2, 0) and (0, 0, 2) Text Transcription: 1/V iint_Q int f(x, y, z) dV

Average Value In Exercises 69-72, find the average value of the function over the given solid. The average value of a continuous function f(x, y, z) over a solid region Q is \(\frac{1}{V} \iint_{Q} \int f(x, y, z) d V\) where V is the volume of the solid region Q. f(x, y, z) = x + y over the solid bounded by the sphere \(x^{2}+y^{2}+z^{2}=3\) Text Transcription: 1/V iint_Q int f(x, y, z) dV x^2 + y^2 + z^2 = 3

Find the solid region Q where the triple integral \(\iiint_{Q}\left(1-2 x^{2}-y^{2}-3 z^{2}\right) d V\) is a maximum. Use a computer algebra system to approximate the maximum value. What is the exact maximum value? Text Transcription: iiint_Q (1 - 2x^2 - y^2 - 3z^2) dV

Find the solid region Q where the triple integral \(\iiint_{Q}\left(1-x^{2}-y^{2}-z^{2}\right) d V\) is a maximum. Use a computer algebra system to approximate the maximum value. What is the exact maximum value? Text Transcription: iiint_Q (1 - x^2 - y^2 - z^2) dV

Solve for a in the triple integral. \(\int_{0}^{1} \int_{0}^{3-a-y^{2}} \int_{a}^{4-x-y^{2}} d z d x d y=\frac{14}{15}\) Text Transcription: int_0 ^1 int_0 ^3 - a - y^2 int_a ^4 - x - y^2 dz dx dy = 14/15

Determine the value of b such that the volume of the ellipsoid \(x^{2}+\left(y^{2} / b^{2}\right)+\left(z^{2} / 9\right)=1\) is \(16 \pi\). Text Transcription: x^2 + (y^2 / b^2) + (z^2 / 9) = 1 16 pi