In Exercises 31 and 32, sketch a graph of the region

In Exercises 1 and 2, evaluate the integral. \(\int_{0}^{2 x} x y^{3} d y\) Text Transcription: int_{0}^{2x} xy^3 dy

In Exercises 1 and 2, evaluate the integral. \(\int_{y}^{2 y}\left(x^{2}+y^{2}\right) d x\) Text Transcription: int_{y}^{2y} (x^2 + y^2) dx

In Exercises 3 - 6, evaluate the iterated integral. \(\int_{0}^{1} \int_{0}^{1+x}(3 x+2 y) d y d x\) Text Transcription: int_{0}^{1} int_{0}^{1 + x} (3x + 2y) dy dx

In Exercises 3 - 6, evaluate the iterated integral. \(\int_{0}^{2} \int_{x^{2}}^{2 x}\left(x^{2}+2 y\right) d y d x\) Text Transcription: int_{0}^{2} int_{x^2}^{2x} (x^2 + 2y) dy dx

In Exercises 3 - 6, evaluate the iterated integral. \(\int_{0}^{3} \int_{0}^{\sqrt{9-x^{2}}} 4 x d y d x\) Text Transcription: int_{0}^{3} int_{0}^{sqrt{9 - x^2} 4x dy dx

In Exercises 3 - 6, evaluate the iterated integral. \(\int_{0}^{1} \int_{0}^{2 y}\left(9+3 x^{2}+3 y^{2}\right) d x d y\) Text Transcription: int_{0}^{1} int_{0}^{2y} (9 + 3x^2 + 3y^2) dx dy

In Exercises 7 - 10, use an iterated integral to find the area of the region bounded by the graphs of the equations. x + 3y = 3, x = 0, y = 0

In Exercises 7 - 10, use an iterated integral to find the area of the region bounded by the graphs of the equations. \(y=6 x-x^{2}, y=x^{2}-2 x\) Text Transcription: y = 6x - x^2, y = x^2 - 2x

In Exercises 7 - 10, use an iterated integral to find the area of the region bounded by the graphs of the equations. y = x, y = 2x + 2, x = 0, x = 4

In Exercises 7 - 10, use an iterated integral to find the area of the region bounded by the graphs of the equations. \(x=y^{2}+1, x=0, y=0, y=2\) Text Transcription: x = y^2 + 1, x = 0, y = 0, y = 2

In Exercises 11 - 14, sketch the region R whose area is given by the iterated integral. Then switch the order of integration and show that both orders yield the same area. \(\int_{2}^{4} \int_{1}^{5} d x d y\) Text Transcription: int_{2}^{4} int_{1}^{5} dx dy

In Exercises 11 - 14, sketch the region R whose area is given by the iterated integral. Then switch the order of integration and show that both orders yield the same area. \(\int_{0}^{3} \int_{0}^{x} d y d x+\int_{3}^{6} \int_{0}^{6-x} d y d x\) Text Transcription: int_{0}^{3} int_{0}^{x} dy dx + int_{3}^{6} int_{0}^{6 - x} dy dx

In Exercises 11 - 14, sketch the region R whose area is given by the iterated integral. Then switch the order of integration and show that both orders yield the same area. \(\int_{0}^{4} \int_{2 x}^{8} d y d x\) Text Transcription: int_{0}^{4} int_{2x}^8 dy dx

In Exercises 11 - 14, sketch the region R whose area is given by the iterated integral. Then switch the order of integration and show that both orders yield the same area. \(\int_{-3}^{3} \int_{0}^{9-y^{2}} d x d y\) Text Transcription: int_{-3}^{3} int_{0}^{9 - y^2} dx dy

In Exercises 15 and 16, set up integrals for both orders of integration. Use the more convenient order to evaluate the integral over the region R. \(\int_{R} \int 4 x y d A\) R: rectangle with vertices (0, 0), (0, 4), (2, 4), (2, 0) Text Transcription: int_R int 4xy dA

In Exercises 15 and 16, set up integrals for both orders of integration. Use the more convenient order to evaluate the integral over the region R. \(\int_{R} \int 6 x^{2} d A\) R: region bounded by \(y=0, y=\sqrt{x}, x=1\) Text Transcription: int_R int 6x^2 dA y =0, y = sqrt{x}, x = 1

In Exercises 17 – 20, use a double integral to find the volume of the indicated solid.

In Exercises 17 – 20, use a double integral to find the volume of the indicated solid.

In Exercises 17 – 20, use a double integral to find the volume of the indicated solid.

In Exercises 17 – 20, use a double integral to find the volume of the indicated solid.

In Exercises 21 and 22, find the average value of f(x, y) over the plane region R. \(f(x)=16-x^{2}-y^{2}\) R: rectangle with vertices (2,2), (-2,2), (-2,-2), (2,-2) Text Transcription: f(x) = 16 - x^2 - y^2

In Exercises 21 and 22, find the average value of f(x, y) over the plane region R. \(f(x)=2 x^{2}+y^{2}\) R: square with vertices (0, 0), (3, 0), (3, 3), (0, 3) Text Transcription: f(x) = 2x^2 + y^2

The temperature in degrees Celsius on the surface of a metal plate is \(T(x, y)=40-6 x^{2}-y^{2}\) where x and y are measured in centimeters. Estimate the average temperature when x varies between 0 and 3 centimeters and y varies between 0 and 5 centimeters. Text Transcription: T(x, y) = 40 - 6x^2 - y^2

A firm's profit P from marketing two soft drinks is \(P=192 x+576 y-x^{2}-5 y^{2}-2 x y-5000\) where x and y represent the numbers of units of the two soft drinks. Estimate the average weekly profit when x varies between 40 and 50 units and y varies between 45 and 60 units. Text Transcription: P = 192x + 576y - x^2 - 5y^2 - 2xy - 5000

In Exercises 25 and 26, evaluate the iterated integral by converting to polar coordinates. \(\int_{0}^{h} \int_{0}^{x} \sqrt{x^{2}+y^{2}} d y d x\) Text Transcription: int_{0}^{h} int_{0}^{x} sqrt{x^2 + y^2} dy dx

In Exercises 25 and 26, evaluate the iterated integral by converting to polar coordinates. \(\int_{0}^{4} \int_{0}^{\sqrt{16-y^{2}}}\left(x^{2}+y^{2}\right) d x d y\) Text Transcription: int_{0}^{4} int_{0}^{sqrt{16 - y^2} (x^2 + y^2) dx dy

In Exercises 27 and 28, use a double integral in polar coordinates to find the volume of the solid bounded by the graphs of the equations. \(z=x y^{2}, x^{2}+y^{2}=9\), first octant Text Transcription: z = xy^2, x^2 + y^2 = 9

In Exercises 27 and 28, use a double integral in polar coordinates to find the volume of the solid bounded by the graphs of the equations. \(z=\sqrt{25-x^{2}-y^{2}}, z=0, x^{2}+y^{2}=16\) Text Transcription: z = sqrt{25 - x^2 - y^2}, z = 0, x^2 + y^2 = 16

In Exercises 29 and 30, use a double integral to find the area of the shaded region.

In Exercises 29 and 30, use a double integral to find the area of the shaded region.

In Exercises 31 and 32, sketch a graph of the region bounded by the graphs of the equations. Then use a double integral to find the area of the region. Inside the cardioid \(r=2+2 \cos \theta\) and outside the circle r = 3 Text Transcription: r = 2 + 2 cos theta

In Exercises 31 and 32, sketch a graph of the region bounded by the graphs of the equations. Then use a double integral to find the area of the region. Inside the circle \(r=3 \sin \theta\) and outside the cardioid \(r=1+\sin \theta\) Text Transcription: r = 3 sin theta r = 1 + sin theta

Consider the region R in the xy-plane bounded by the graph of the equation \(\left(x^{2}+y^{2}\right)^{2}=9\left(x^{2}-y^{2}\right)\) (a) Convert the equation to polar coordinates. Use a graphing utility to graph the equation. (b) Use a double integral to find the area of the region R. (c) Use a computer algebra system to determine the volume of the solid over the region R and beneath the hemisphere \(z=\sqrt{9-x^{2}-y^{2}}\). Text Transcription: (x^2 + y^2)^2 = 9(x^2 - y^2) z = sqrt{9 - x^2 - y^2}

Combine the sum of the two iterated integrals into a single iterated integral by converting to polar coordinates. Evaluate the resulting iterated integral. \(\int_{0}^{8 / \sqrt{13}} \int_{0}^{3 x / 2} x y d y d x+\int_{8 / \sqrt{13}}^{4} \int_{0}^{\sqrt{16-x^{2}}} x y d y d x\) Text Transcription: int_{0}^{8/sqrt{13}} int_{0}^{3x/2} xy dy dx + int_{8/sqrt{13}}^{4} int_{0}^{sqrt{16 - x^2} xy dy dx

In Exercises 35 - 38, find the mass and center of mass of the lamina bounded by the graphs of the equations for the given density. (Hint: Some of the integrals are simpler in polar coordinates.) \(y=x^{3}, y=0, x=2, \rho=k x\) Text Transcription: y = x^3, y = 0, x = 2, rho = kx

In Exercises 35 - 38, find the mass and center of mass of the lamina bounded by the graphs of the equations for the given density. (Hint: Some of the integrals are simpler in polar coordinates.) \(y=\frac{2}{x}, y=0, x=1, x=2, \rho=k y\) Text Transcription: y = 2 / x, y = 0, x = 1, x = 2, rho = ky

In Exercises 35 - 38, find the mass and center of mass of the lamina bounded by the graphs of the equations for the given density. (Hint: Some of the integrals are simpler in polar coordinates.) \(y=2 x, y=2 x^{3}, x \geq 0, y \geq 0, \rho=k x y\) Text Transcription: y = 2x, y = 2x^3, x geq 0, y geq 0, rho = kxy

In Exercises 35 - 38, find the mass and center of mass of the lamina bounded by the graphs of the equations for the given density. (Hint: Some of the integrals are simpler in polar coordinates.) \(y=6-x, y=0, x=0, \rho=k x^{2}\) Text Transcription: y = 6 - x, y = 0, x = 0, rho = kx^2

In Exercises 39 and 40 , find \(I_{x}, I_{y}, I_{0}, \overline{\bar{x}}\), and \(\overline{\bar{y}}\) for the lamina bounded by the graphs of the equations. \(y=0, y=b, x=0, x=a, \rho=k x\) Text Transcription: I_x, I_y, I_0, overline bar x overline bar y y = 0, y = b, x = 0, x = a, rho = kx

In Exercises 39 and 40 , find \(I_{x}, I_{y}, I_{0}, \overline{\bar{x}}\), and \(\overline{\bar{y}}\) for the lamina bounded by the graphs of the equations. \(y=4-x^{2}, y=0, x>0, \rho=k y\) Text Transcription: I_x, I_y, I_0, overline bar x overline bar y y = 4 - x^2, y = 0, x > 0, rho = ky

In Exercises 41 - 44, find the area of the surface given by z = f(x, y) over the region R. (Hint: Some of the integrals are simpler in polar coordinates.) \(f(x, y)=25-x^{2}-y^{2}\) \(R=\left\{(x, y): x^{2}+y^{2} \leq 25\right\}\) Text Transcription: f(x, y) = 25 - x^2 - y^2 R = {(x, y): x^2 + y^2 leq 25}

In Exercises 41 - 44, find the area of the surface given by z = f(x, y) over the region R. (Hint: Some of the integrals are simpler in polar coordinates.) f(x, y) = 8 + 4x - 5y \(R=\left\{(x, y): x^{2}+y^{2} \leq 1\right\}\) Text Transcription: R = {(x, y): x^2 + y^2 leq 1}

In Exercises 41 - 44, find the area of the surface given by z = f(x, y) over the region R. (Hint: Some of the integrals are simpler in polar coordinates.) \(f(x, y)=9-y^{2}\) R: triangle bounded by the graphs of the equations y = x, y = -x, and y = 3 Text Transcription: f(x, y) = 9 - y^2

In Exercises 41 - 44, find the area of the surface given by z = f(x, y) over the region R. (Hint: Some of the integrals are simpler in polar coordinates.) \(f(x, y)=4-x^{2}\) R: triangle bounded by the graphs of the equations y = x, y = -x, and y = 2 Text Transcription: f(x, y) = 4 - x^2

A new auditorium is built with a foundation in the shape of one-fourth of a circle of radius 50 feet. So, it forms a region R bounded by the graph of \(x^{2}+y^{2}=50^{2}\) with \(x \geq 0\) and \(y \geq 0\). The following equations are models for the floor and ceiling. Floor: \(z=\frac{x+y}{5}\) Ceiling: \(z=20+\frac{x y}{100}\) (a) Calculate the volume of the room, which is needed to determine the heating and cooling requirements. (b) Find the surface area of the ceiling. Text Transcription: x^2 + y^2 = 50^2 x geq 0 y geq 0 z = x + y / 5 z = 20 + xy / 100

The roof over the stage of an open-air theater at a theme park is modeled by \(f(x, y)=25\left[1+e^{-\left(x^{2}+y^{2}\right) / 1000} \cos ^{2}\left(\frac{x^{2}+y^{2}}{1000}\right)\right]\) where the stage is a semicircle bounded by the graphs of \(y=\sqrt{50^{2}-x^{2}}\) and y = 0. (a) Use a computer algebra system to graph the surface. (b) Use a computer algebra system to approximate the number of square feet of roofing required to cover the surface. Text Transcription: f(x, y) = 25 [1 + e^{-(x^2 + y^2) / 1000} cos^2 (x^2 + y^2 / 1000)] y = sqrt{50^2 - x^2}

In Exercises 47 – 50, evaluate the triple iterated integral. \(\int_{0}^{4} \int_{0}^{1} \int_{0}^{2}(2 x+y+4 z) d y d z d x\) Text Transcription: int_{0}^{4} int_{0}^{1} int_{0}^{2}(2x + y + 4z) dy dz dx

In Exercises 47 – 50, evaluate the triple iterated integral. \(\int_{0}^{2} \int_{0}^{y} \int_{0}^{x y} y d z d x d y\) Text Transcription: int_{0}^{2} int_{0}^{y} int_{0}^{xy} y dz dx dy

In Exercises 47 – 50, evaluate the triple iterated integral. \(\int_{0}^{a} \int_{0}^{b} \int_{0}^{c}\left(x^{2}+y^{2}+z^{2}\right) d x d y d z\) Text Transcription: int_{0}^{a} int_{0}^{b} int_{0}^{c} (x^2 + y^2 + z^2) dx dy dz

In Exercises 47 – 50, evaluate the triple iterated integral. \(\int_{0}^{3} \int_{\pi / 2}^{\pi} \int_{2}^{5} z \sin x d y d x d z\) Text Transcription: int_{0}^{3} int_{pi / 2}^{pi} int_{2}^{5} z sin x dy dx dz

In Exercises 51 and 52, use a computer algebra system to approximate the iterated integral. \(\int_{-1}^{1} \int_{-\sqrt{1-x^{2}}}^{\sqrt{1-x^{2}}} \int_{-\sqrt{1-x^{2}-y^{2}}}^{\sqrt{1-x^{2}-y^{2}}}\left(x^{2}+y^{2}\right) d z d y d x\) Text Transcription: int_{-1}^{1} int_{-sqrt{1 - x^2}}^{sqrt{1 - x^2}} int_{-sqrt{1 - x^2 - y^2}}^{sqrt{1 - x^2 -y^2}}(x^2 + y^2) dz dy dx

In Exercises 51 and 52, use a computer algebra system to approximate the iterated integral. \(\int_{0}^{2} \int_{0}^{\sqrt{4-x^{2}}} \int_{0}^{\sqrt{4-x^{2}-y^{2}}} x y z d z d y d x\) Text Transcription: int_{0}^{2} int_{0}^{sqrt{4 - x^2}} int_{0}^{sqrt{4 - x^2 - y^2}} xyz dz dy dx

In Exercises 53 and 54, use a triple integral to find the volume of the solid bounded by the graphs of the equations. \(z=x y, z=0,0 \leq x \leq 3,0 \leq y \leq 4\) Text Transcription: z = xy, z = 0, 0 leq x leq 3, 0 leq y leq 4

In Exercises 53 and 54, use a triple integral to find the volume of the solid bounded by the graphs of the equations. z = 8 - x - y, z = 0, y = x, y = 3, x = 0

In Exercises 55 and 56, sketch the solid whose volume is given by the iterated integral and rewrite the integral using the indicated order of integration. \(\int_{0}^{1} \int_{0}^{y} \int_{0}^{\sqrt{1-x^{2}}} d z d x d y\) Rewrite using the order dz dy dx. Text Transcription: int_{0}^{1} int_{0}^{y} int_{0}^{sqrt{1 - x^2}} dz dx dy

In Exercises 55 and 56, sketch the solid whose volume is given by the iterated integral and rewrite the integral using the indicated order of integration. \(\int_{0}^{6} \int_{0}^{6-x} \int_{0}^{6-x-y} d z d y d x\) Rewrite using the order dy dx dz. Text Transcription: int_{0}^{6} int_{0}^{6 - x} int_{0}^{6 - x - y} dz dy dx

In Exercises 57 and 58, find the mass and the indicated coordinates of the center of mass of the solid region Q of density \(\rho\) bounded by the graphs of the equations. Find \(\bar{x}\) using \(\rho(x, y, z)=k\). Q: x + y + z = 10, x = 0, y = 0, z = 0 Text Transcription: rho bar x rho(x, y, z) = k

In Exercises 57 and 58, find the mass and the indicated coordinates of the center of mass of the solid region Q of density \(\rho\) bounded by the graphs of the equations. Find \(\bar{y}\) using \(\rho(x, y, z)=k x\). Q: z = 5 - y, z = 0, y = 0, x = 0, x = 5 Text Transcription: rho bar y rho(x, y, z) = kx

In Exercises 59 - 62, evaluate the iterated integral. \(\int_{0}^{3} \int_{0}^{\pi / 3} \int_{0}^{4} r \cos \theta d r d \theta d z\) Text Transcription: int_{0}^{3} int_{0}^{pi/3} int_{0}^{4} r cos theta dr d theta dz

In Exercises 59 - 62, evaluate the iterated integral. \(\int_{0}^{\pi / 2} \int_{0}^{3} \int_{0}^{4-z} z d r d z d \theta\) Text Transcription: int_{0}^{pi/2} int_{0}^{3} int_{0}^{4 - z} z dr dz d theta

In Exercises 59 - 62, evaluate the iterated integral. \(\int_{0}^{\pi / 2} \int_{0}^{\pi / 2} \int_{0}^{2} \rho^{2} d \rho d \theta d \phi\) Text Transcription: int_{0}^{pi/2} int_{0}^{pi/2} int_{0}^{2} rho^{2} d rho d theta d phi

In Exercises 59 - 62, evaluate the iterated integral. \(\int_{0}^{\pi / 4} \int_{0}^{\pi / 4} \int_{0}^{\cos \phi} \cos \theta d \rho d \phi d \theta\) Text Transcription: int_{0}^{pi/4} int_{0}^{pi/4} int_{0}^{cos phi} cos theta d rho d phi d theta

In Exercises 63 and 64, use a computer algebra system to approximate the iterated integral. \(\int_{0}^{\pi} \int_{0}^{2} \int_{0}^{3} \sqrt{z^{2}+4} d z d r d \theta\) Text Transcription: int_{0}^{pi} int_{0}^{2} int_{0}^{3} sqrt{z^2 + 4} dz dr d theta

In Exercises 63 and 64, use a computer algebra system to approximate the iterated integral. \(\int_{0}^{\pi / 2} \int_{0}^{\pi / 2} \int_{0}^{\cos \phi} \rho^{2} \cos \theta d \rho d \theta d \phi\) Text Transcription: int_{0}^{pi/2} int_{0}^{pi/2} int_{0}^{cos phi} rho^2 cos theta d rho d theta d phi

Use cylindrical coordinates to find the volume of the solid bounded above by \(z=8-x^{2}-y^{2}\) and below by \(z=x^{2}+y^{2}\). Text Transcription: z = 8 - x^2 - y^2 z = x^2 + y^2.

Use spherical coordinates to find the volume of the solid bounded above by \(x^{2}+y^{2}+z^{2}=36\) and below by \(z=\sqrt{x^{2}+y^{2}}\). Text Transcription: x^2 + y^2 + z^2 = 36 z = sqrt{x^2 + y^2}

Finding a Jacobian In Exercises 67 - 70, find the Jacobian \(\partial(x, y) / \partial(u, v)\) for the indicated change of variables. x = u + 3v, y = 2u - 3v Text Transcription: partial(x, y)/partial(u, v)

Finding a Jacobian In Exercises 67 - 70, find the Jacobian \(\partial(x, y) / \partial(u, v)\) for the indicated change of variables. \(x=u^{2}+v^{2}, y=u^{2}-v^{2}\) Text Transcription: partial(x, y)/partial(u, v) x = u^2 + v^2, y = u^2 - v^2

Finding a Jacobian In Exercises 67 - 70, find the Jacobian \(\partial(x, y) / \partial(u, v)\) for the indicated change of variables. \(x=u \sin \theta+v \cos \theta, y=u \cos \theta+v \sin \theta\) Text Transcription: partial(x, y)/partial(u, v) x = u sin theta + v cos theta, y = u cos theta + v sin theta

Finding a Jacobian In Exercises 67 - 70, find the Jacobian \(\partial(x, y) / \partial(u, v)\) for the indicated change of variables. \(x=u v, y=\frac{v}{u}\) Text Transcription: partial(x, y)/partial(u, v) x = uv, y = v / u

In Exercises 71 - 74, use the indicated change of variables to evaluate the double integral. \(\int_{R} \int \ln (x+y) d A\) \(x=\frac{1}{2}(u+v), y=\frac{1}{2}(u-v)\) Text Transcription: int_R int ln (x + y) dA x = 1 / 2 (u + v), y = 1 / 2 (u - v)

In Exercises 71 - 74, use the indicated change of variables to evaluate the double integral. \(\int_{R} \int 16 x y d A\) \(x=\frac{1}{4}(u+v), y=\frac{1}{2}(v-u)\) Text Transcription: int_R int 16xy dA x = 1 / 4 (u + v), y = 1 / 2 (v - u)

In Exercises 71 - 74, use the indicated change of variables to evaluate the double integral. \(\int_{R} \int\left(x y+x^{2}\right) d A\) \(x=u, y=\frac{1}{3}(u-v)\) Text Transcription: int_R int (xy + x^2) dA x = u, y = 1 / 3 (u - v)

In Exercises 71 - 74, use the indicated change of variables to evaluate the double integral. \(\int_{R} \int \frac{x}{1+x^{2} y^{2}} d A\) \(x=u, y=\frac{v}{u}\) Text Transcription: int_R int x / 1 + x^{2} y^{2} dA x = u, y = v / u