Solved: Evaluating a Double Integral Using a Change of Variables In Exercises 1520, use

In Exercises 1 - 8, find the Jacobian \(\partial(x, y) / \partial(u, v)\) for the indicated change of variables. \(x=-\frac{1}{2}(u-v), y=\frac{1}{2}(u+v)\) Text Transcription: partial(x, y)/partial(u, v) x = - 1 / 2 (u - v), y = 1 / 2 (u + v)

In Exercises 1 - 8, find the Jacobian \(\partial(x, y) / \partial(u, v)\) for the indicated change of variables. x = au + bv, y = cu + dv Text Transcription: partial(x, y)/partial(u, v)

In Exercises 1 - 8, find the Jacobian \(\partial(x, y) / \partial(u, v)\) for the indicated change of variables. \(x=u-v^{2}, y=u+v\) Text Transcription: partial(x, y)/partial(u, v) x = u - v^2, y = u + v

In Exercises 1 - 8, find the Jacobian \(\partial(x, y) / \partial(u, v)\) for the indicated change of variables. x = uv - 2u, y = uv Text Transcription: partial(x, y)/partial(u, v)

In Exercises 1 - 8, find the Jacobian \(\partial(x, y) / \partial(u, v)\) for the indicated change of variables. \(x=u \cos \theta-v \sin \theta, y=u \sin \theta+v \cos \theta\) Text Transcription: partial(x, y)/partial(u, v) x = u cos theta - v sin theta, y = u sin theta + v cos theta

In Exercises 1 - 8, find the Jacobian \(\partial(x, y) / \partial(u, v)\) for the indicated change of variables. x = u + a, y = v + a Text Transcription: partial(x, y)/partial(u, v)

In Exercises 1 - 8, find the Jacobian \(\partial(x, y) / \partial(u, v)\) for the indicated change of variables. \(x=e^{u} \sin v, y=e^{u} \cos v\) Text Transcription: partial(x, y)/partial(u, v) x = e^u sin v, y = e^u cos v

In Exercises 1 - 8, find the Jacobian \(\partial(x, y) / \partial(u, v)\) for the indicated change of variables. x = u/v, y = u + v Text Transcription: partial(x, y)/partial(u, v)

In Exercises 9 - 12, sketch the image S in the uv-plane of the region R in the xy-plane using the given transformations. x = 3u + 2v y = 3v

In Exercises 9 - 12, sketch the image S in the uv-plane of the region R in the xy-plane using the given transformations. \(x=\frac{1}{3}(4 u-v)\) \(y=\frac{1}{3}(u-v)\) Text Transcription: x = 1 / 3 (4u - v) y = 1 / 3 (u - v)

In Exercises 9 - 12, sketch the image S in the uv-plane of the region R in the xy-plane using the given transformations. \(x=\frac{1}{2}(u+v)\) \(y=\frac{1}{2}(u-v)\) Text Transcription: x = 1 / 2 (u + v) y = 1 / 2 (u - v)

In Exercises 9 - 12, sketch the image S in the uv-plane of the region R in the xy-plane using the given transformations. \(x=\frac{1}{3}(v-u)\) \(y=\frac{1}{3}(2 v+u)\) Text Transcription: x = 1 / 3 (v - u) y = 1 / 3(2v + u)

In Exercises 13 and 14, verify the result of the indicated example by setting up the integral using dy dx or dx dy for dA. Then use a computer algebra system to evaluate the integral. Example 3

In Exercises 13 and 14, verify the result of the indicated example by setting up the integral using dy dx or dx dy for dA. Then use a computer algebra system to evaluate the integral. Example 4

In Exercises 15 - 20, use the indicated change of variables to evaluate the double integral. \(\int_{R} \int 4\left(x^{2}+y^{2}\right) d A\) \(x=\frac{1}{2}(u+v)\) \(y=\frac{1}{2}(u-v)\) Text Transcription: int_R int 4 (x^2 + y^2) dA x = 1 / 2 (u + v) y = 1 / 2 (u - v)

In Exercises 15 - 20, use the indicated change of variables to evaluate the double integral. \(\int_{R} \int 60 x y d A\) \(x=\frac{1}{2}(u+v)\) \(y=-\frac{1}{2}(u-v)\) Text Transcription: int_{R} int 60xy dA x = 1 / 2 (u + v) y = -1 / 2 (u - v)

In Exercises 15 - 20, use the indicated change of variables to evaluate the double integral. \(\int_{R} \int y(x-y) d A\) x = u + v y = u Text Transcription: int_{R} int y(x - y) dA

In Exercises 15 - 20, use the indicated change of variables to evaluate the double integral. \(\int_{R} \int 4(x+y) e^{x-y} d A\) \(x=\frac{1}{2}(u+v)\) \(y=\frac{1}{2}(u-v)\) Text Transcription: int_R int 4(x + y) e^{x - y} dA x = 1 / 2 (u + v) y = 1 / 2 (u - v)

In Exercises 15 - 20, use the indicated change of variables to evaluate the double integral. \(\int_{R} \int e^{-x y / 2} d A\) \(x=\sqrt{\frac{v}{u}}, y=\sqrt{u v}\) Text Transcription: int_R int e^{-xy/2} dA x = sqrt{v / u}, y = sqrt{uv}

In Exercises 15 - 20, use the indicated change of variables to evaluate the double integral. \(\int_{R} \int y \sin x y d A\) \(x=\frac{u}{v}, \quad y=v\) Text Transcription: int_R int y sin xy dA x = u / v, y = v

In Exercises 21 - 28, use a change of variables to find the volume of the solid region lying below the surface z = f(x, y) and above the plane region R. f(x, y) = 48xy R: region bounded by the square with vertices (1, 0), (0, 1), (1, 2), (2, 1)

In Exercises 21 - 28, use a change of variables to find the volume of the solid region lying below the surface z = f(x, y) and above the plane region R. \(f(x, y)=(3 x+2 y)^{2} \sqrt{2 y-x}\) R: region bounded by the parallelogram with vertices (0, 0), (-2, 3), (2, 5), (4, 2) Text Transcription: f(x, y) = (3x + 2y)^{2} sqrt{2y - x}

In Exercises 21 - 28, use a change of variables to find the volume of the solid region lying below the surface z = f(x, y) and above the plane region R. \(f(x, y)=(x+y) e^{x-y}\) R: region bounded by the square with vertices (4, 0), (6, 2), (4, 4), (2, 2) Text Transcription: f(x, y) = (x + y) e^{x - y}

In Exercises 21 - 28, use a change of variables to find the volume of the solid region lying below the surface z = f(x, y) and above the plane region R. \(f(x, y)=(x+y)^{2} \sin ^{2}(x-y)\) R: region bounded by the square with vertices \((\pi, 0), (3 \pi / 2, \pi / 2),(\pi, \pi),(\pi / 2, \pi / 2)\) Text Transcription: f(x, y) = (x + y)^2 sin^2 (x - y) (pi, 0), (3 pi/2, pi/2), (pi, pi),(pi/2, pi/2)

In Exercises 21 - 28, use a change of variables to find the volume of the solid region lying below the surface z = f(x, y) and above the plane region R. \(f(x, y)=\sqrt{(x-y)(x+4 y)}\) R: region bounded by the parallelogram with vertices (0, 0), (1, 1), (5, 0), (4, -1) Text Transcription: f(x, y) = sqrt{(x - y)(x + 4y)}

In Exercises 21 - 28, use a change of variables to find the volume of the solid region lying below the surface z = f(x, y) and above the plane region R. \(f(x, y)=(3 x+2 y)(2 y-x)^{3 / 2}\) R: region bounded by the parallelogram with vertices (0, 0), (-2, 3), (2, 5), (4, 2) Text Transcription: f(x, y) = (3x + 2y) (2y - x)^{3/2}

In Exercises 21 - 28, use a change of variables to find the volume of the solid region lying below the surface z = f(x, y) and above the plane region R. \(f(x, y)=\sqrt{x+y}\) R: region bounded by the triangle with vertices (0, 0), (a, 0), (0, a), where a > 0 Text Transcription: f(x, y) = sqrt{x + y}

In Exercises 21 - 28, use a change of variables to find the volume of the solid region lying below the surface z = f(x, y) and above the plane region R. \(f(x, y)=\frac{x y}{1+x^{2} y^{2}}\) R: region bounded by the graphs of xy = 1, xy = 4, x = 1, x = 4 (Hint: Let x = u, y = v/u.) Text Transcription: f(x, y) = xy / 1 + x^{2} y^{2}

The substitutions u = 2x - y and v = x + y make the region R (see figure) into a simpler region S in the uv-plane. Determine the total number of sides of S that are parallel to either the u-axis or the v-axis.

The region R is transformed into a simpler region S (see figure). Which substitution can be used to make the transformation? (a) u =3y - x, v = y - x (b) u = y - x, v = 3y - x

Consider the region R in the xy-plane bounded by the ellipse \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\) and the transformations x = au and y = bv. (a) Sketch the graph of the region R and its image S under the given transformation. (b) Find \(\frac{\partial(x, y)}{\partial(u, v)}\). (c) Find the area of the ellipse. Text Transcription: x^2 / a^2 + y^2 / b^2 = 1 partial(x, y) / partial(u, v)

Use the result of Exercise 31 to find the volume of each dome-shaped solid lying below the surface z = f(x, y) and above the elliptical region R. (Hint: After making the change of variables given by the results in Exercise 31, make a second change of variables to polar coordinates.) (a) \(f(x, y)=16-x^{2}-y^{2}\) \(R: \frac{x^{2}}{16}+\frac{y^{2}}{9} \leq 1\) (b) \(f(x, y)=A \cos \left(\frac{\pi}{2} \sqrt{\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}}\right)\) \(R: \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}} \leq 1\) Text Transcription: f(x, y) = 16 - x^2 - y^2 R: x^2 / 16 + y^2 / 9 leq 1 f(x, y) = A cos (pi / 2 sqrt{x^2 / a^2} + y^2 / b^2) R: x^2 / a^2 + y^2 / b^2 leq 1

Jacobian State the definition of the Jacobian.

Change of Variables Describe how to use the Jacobian to change variables in double integrals.

In Exercises 35 - 40, find the Jacobian \(\frac{\partial(x, y, z)}{\partial(u, v, w)}\) for the indicated change of variables. If x = f(u, v, w), y = g(u, v, w), and z = h(u, v, w) then the Jacobian of x, y, and z with respect to u, v, and w is \(\frac{\partial(x, y, z)}{\partial(u, v, w)}=\left|\begin{array}{lll}\frac{\partial x}{\partial u} & \frac{\partial x}{\partial v} & \frac{\partial x}{\partial w} \\ \frac{\partial y}{\partial u} & \frac{\partial y}{\partial v} & \frac{\partial y}{\partial w} \\ \frac{\partial z}{\partial u} & \frac{\partial z}{\partial v} & \frac{\partial z}{\partial w}\end{array}\right|\) x = u(1 - v), y = uv(1 - w), z = uvw Text Transcription: partial(x, y, z) / partial(u, v, w) partial(x, y, z) / partial(u, v, w)} = |partial x / partial u partial x / partial v partial x partial w partial y / partial u partial y / partial v partial y / partial w partial z / partial u partial z / partial v partial z / partial w|

In Exercises 35 - 40, find the Jacobian \(\frac{\partial(x, y, z)}{\partial(u, v, w)}\) for the indicated change of variables. If x = f(u, v, w), y = g(u, v, w), and z = h(u, v, w) then the Jacobian of x, y, and z with respect to u, v, and w is \(\frac{\partial(x, y, z)}{\partial(u, v, w)}=\left|\begin{array}{lll}\frac{\partial x}{\partial u} & \frac{\partial x}{\partial v} & \frac{\partial x}{\partial w} \\ \frac{\partial y}{\partial u} & \frac{\partial y}{\partial v} & \frac{\partial y}{\partial w} \\ \frac{\partial z}{\partial u} & \frac{\partial z}{\partial v} & \frac{\partial z}{\partial w}\end{array}\right|\) x = 4u - v, y = 4v - w, z = u + w Text Transcription: partial(x, y, z) / partial(u, v, w) partial(x, y, z) / partial(u, v, w)} = |partial x / partial u partial x / partial v partial x partial w partial y / partial u partial y / partial v partial y / partial w partial z / partial u partial z / partial v partial z / partial w|

In Exercises 35 - 40, find the Jacobian \(\frac{\partial(x, y, z)}{\partial(u, v, w)}\) for the indicated change of variables. If x = f(u, v, w), y = g(u, v, w), and z = h(u, v, w) then the Jacobian of x, y, and z with respect to u, v, and w is \(\frac{\partial(x, y, z)}{\partial(u, v, w)}=\left|\begin{array}{lll}\frac{\partial x}{\partial u} & \frac{\partial x}{\partial v} & \frac{\partial x}{\partial w} \\ \frac{\partial y}{\partial u} & \frac{\partial y}{\partial v} & \frac{\partial y}{\partial w} \\ \frac{\partial z}{\partial u} & \frac{\partial z}{\partial v} & \frac{\partial z}{\partial w}\end{array}\right|\) \(x=\frac{1}{2}(u+v), y=\frac{1}{2}(u-v), z=2 u v w\) Text Transcription: partial(x, y, z) / partial(u, v, w) partial(x, y, z) / partial(u, v, w)} = |partial x / partial u partial x / partial v partial x partial w partial y / partial u partial y / partial v partial y / partial w partial z / partial u partial z / partial v partial z / partial w| x = 1 / 2 (u + v), y = 1 / 2 (u - v), z = 2uvw

In Exercises 35 - 40, find the Jacobian \(\frac{\partial(x, y, z)}{\partial(u, v, w)}\) for the indicated change of variables. If x = f(u, v, w), y = g(u, v, w), and z = h(u, v, w) then the Jacobian of x, y, and z with respect to u, v, and w is \(\frac{\partial(x, y, z)}{\partial(u, v, w)}=\left|\begin{array}{lll}\frac{\partial x}{\partial u} & \frac{\partial x}{\partial v} & \frac{\partial x}{\partial w} \\ \frac{\partial y}{\partial u} & \frac{\partial y}{\partial v} & \frac{\partial y}{\partial w} \\ \frac{\partial z}{\partial u} & \frac{\partial z}{\partial v} & \frac{\partial z}{\partial w}\end{array}\right|\) x = u - v + w, y = 2uv, z = u + v + w Text Transcription: partial(x, y, z) / partial(u, v, w) partial(x, y, z) / partial(u, v, w)} = |partial x / partial u partial x / partial v partial x partial w partial y / partial u partial y / partial v partial y / partial w partial z / partial u partial z / partial v partial z / partial w|

In Exercises 35 - 40, find the Jacobian \(\frac{\partial(x, y, z)}{\partial(u, v, w)}\) for the indicated change of variables. If x = f(u, v, w), y = g(u, v, w), and z = h(u, v, w) then the Jacobian of x, y, and z with respect to u, v, and w is \(\frac{\partial(x, y, z)}{\partial(u, v, w)}=\left|\begin{array}{lll}\frac{\partial x}{\partial u} & \frac{\partial x}{\partial v} & \frac{\partial x}{\partial w} \\ \frac{\partial y}{\partial u} & \frac{\partial y}{\partial v} & \frac{\partial y}{\partial w} \\ \frac{\partial z}{\partial u} & \frac{\partial z}{\partial v} & \frac{\partial z}{\partial w}\end{array}\right|\) Spherical Coordinates \(x=\rho \sin \phi \cos \theta, y=\rho \sin \phi \sin \theta, z=\rho \cos \phi\) Text Transcription: partial(x, y, z) / partial(u, v, w) partial(x, y, z) / partial(u, v, w)} = |partial x / partial u partial x / partial v partial x partial w partial y / partial u partial y / partial v partial y / partial w partial z / partial u partial z / partial v partial z / partial w| x = rho sin phi cos theta, y = rho sin phi sin theta, z = rho cos phi

In Exercises 35 - 40, find the Jacobian \(\frac{\partial(x, y, z)}{\partial(u, v, w)}\) for the indicated change of variables. If x = f(u, v, w), y = g(u, v, w), and z = h(u, v, w) then the Jacobian of x, y, and z with respect to u, v, and w is \(\frac{\partial(x, y, z)}{\partial(u, v, w)}=\left|\begin{array}{lll}\frac{\partial x}{\partial u} & \frac{\partial x}{\partial v} & \frac{\partial x}{\partial w} \\ \frac{\partial y}{\partial u} & \frac{\partial y}{\partial v} & \frac{\partial y}{\partial w} \\ \frac{\partial z}{\partial u} & \frac{\partial z}{\partial v} & \frac{\partial z}{\partial w}\end{array}\right|\) Cylindrical Coordinates \(x=r \cos \theta, y=r \sin \theta, z=z\) Text Transcription: partial(x, y, z) / partial(u, v, w) partial(x, y, z) / partial(u, v, w)} = |partial x / partial u partial x / partial v partial x partial w partial y / partial u partial y / partial v partial y / partial w partial z / partial u partial z / partial v partial z / partial w| x = r cos theta, y = r sin theta, z = z

Let A be the area of the region in the first quadrant bounded by the line \(y=\frac{1}{2} x\), the x-axis, and the ellipse \(\frac{1}{9} x^{2}+y^{2}=1\). Find the positive number m such that A is equal to the area of the region in the first quadrant bounded by the line y = mx, the y-axis, and the ellipse \(\frac{1}{9} x^{2}+y^{2}=1\) Text Transcription: y = 1 / 2 x 1 / 9 x^2 + y^2 = 1 1 / 9 x^2 + y^2 = 1