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.
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