Question

∬Rsin(y+/xy−x)dA where R denotes the trapezoidal region with vertices (1,0),(2,0), (0,2), and (0,1) is difficult to integrate directly. Instead: (a) Consider the transform u=y−x and v=y+x suggested by the integrand. (b) Find the inverse transform T in the form x=F(u,v)= and y=G(u,v)= (c) Find the image of R. (i) Find the image of the side S1 joining the vertices (1,0) and (2,0) : (ii) Find the image of the side S2 joining the vertices (2,0) and (0,2) : v= for ≤u≤ (iii) Find the image of the side S3 joining the vertices (0,2) and (0,1) : v= for ≤u≤ (iv) Find the image of the side S4 joining the vertices (1,0) and (0,1) : v= for ≤u≤ (d) Find the Jacobian of the transformation (with alphabetical ordering): (e) Write the transformed integral: ∫ab∫cd dudv where a= b= c= and d=

Answer 1

We approach the double integral by applying a change of variables, finding the inverse transform and the image of the given region under this transform, calculating the Jacobian, and then setting up the transformed integral in uv-plane.

Step-by-step explanation:

To evaluate the double integral ∬ R sin(y + 1/xy - x) dA over the trapezoidal region R, we are advised to use a change of variables. The suggested transformations are u = y - x and v = y + x. We need to find the inverse transformations and the image of the region under this change of variables, calculate the Jacobian, and then set up the transformed integral.

(b) The inverse transformation is found by solving the system of equations given by the transformation:
x = (v - u)/2
y = (u + v)/2

(c) To find the image of the region R under the transformation, we trace the images of the given sides of the trapezoid in the uv-plane. For example, the side S1 transforms from points (1,0) and (2,0) in the xy-plane to the line in the uv-plane by substituting these values into the equations u and v.
(i) Image of side S1: v = u + 2 for 1 ≤ u ≤ 2
And similarly for the other sides S2, S3, and S4.

(d) The Jacobian of the transformation is given by the determinant of the matrix of partial derivatives of x and y with respect to u and v, which happens to be -1/2.

(e) The transformed integral is then set up over the new region in the uv-plane with limits a, b, c, and d, determined by the images of the original trapezoidal region's sides.