Chapter 5: Double Integration
Section 5.6: Changing Variables in a Double Integral
Suppose the equations u=ux,y,v=vx,y map a plane region R in the xy-plane to a region R′ in the uv-plane. Suppose further that the mapping is invertible with equations x=xu,v,y=yu,v that map R′ back to R.
Let the Jacobian of the map from R′ to R be given by
Then a double integral over R is related to an equivalent double over R′ by the "formula"
∫∫Rfx,y dA = ∫∫R′fxu,v,yu,v ∂x,y∂u,v dA′
where dA=dx dy or dy dx, and dA′=du dv or dv du.
The Jacobian of the map from R to R′, that is ∂u,v∂x,y, is the reciprocal of the Jacobian ∂x,y∂u,v. This relationship can be exploited in cases where inverting the mapping equations is algebraically tedious. In some such cases, obtaining the reciprocal of ∂u,v∂x,y and expressing it in terms of u and v might involve simpler manipulations than a direct calculation of ∂x,y∂u,v.
Let R be the interior and boundary of the triangle whose vertices are 1,2, 4,9, and 3,5.
Integrate fx,y=2 x+3 y over R, noting that it takes two iterations to cover R.
Make the change of variables u=2 y−3 x−1/5, v=7 x−3 y−1/5 and evaluate the integral of f over the image of R under this change of variables.
Let R be the interior and boundary of the parallelogram formed by the lines x+y=0, x+y=2, 2 y−3 x=0, 2 y−3 x=7.
Integrate fx,y=x+y2 over R, noting that it takes three iterations to cover R.
Make the change of variables u=x+y, v=2 y−3 x and evaluate the integral of f over the image of R under this change of variables.
Let R be the first-quadrant region bounded by the curves C1:x2−y2=1, C2:x2−y2=4, C3:x2+y2=9, C4:x2+y2=16.
Use the change of coordinates u=x2−y2,v=x2+y2 to evaluate the Cartesian integral ∫∫Rx y dA.
Let R be the region bounded by the curves C1:x2−y2=1, C2:x2−y2=4, C3:2 x y=1, C4:2 x y=5.
Express the Cartesian integral ∫∫Rfx,y dA in terms of the coordinates u=x2−y2,v=2 x y.
Let R be the region bounded by the curves C1:16⁢x2⁢y+16⁢y3+8⁢x−88 y=0, C2:144⁢x2⁢y+144⁢y3−24 x−120 y=0, C3:16⁢x2⁢y+16⁢y3−8 x−8 y=0.
Integrate fx,y=x2+y2/y2 over R, noting that it takes two iterations to cover R. Hint: Solve each bounding curve for x=xy and integrate in the order dx dy.
Make the change of variables u=x2+y2, v=x/y and evaluate the integral of f over the image of R under this change of variables.
<< Previous Section Table of Contents Next Section >>
© Maplesoft, a division of Waterloo Maple Inc., 2023. All rights reserved. This product is protected by copyright and distributed under licenses restricting its use, copying, distribution, and decompilation.
For more information on Maplesoft products and services, visit www.maplesoft.com
Download Help Document