Question

Evaluate the integral by making an appropriate change of variables. 7(x y ex2 ? y2 da, r where r is the rectangle enclosed by the lines x ? y = 0, x ? y = 10, x y = 0, and x y = 4

Answer 1

Take

$\begin{cases}u=x-y\\v=x+y\end{cases}$

so that

$\begin{cases}\mathbf x(u,v)=\dfrac{u+v}2\\\\\mathbf y(u,v)=\dfrac{-u+v}2\end{cases}$

and the Jacobian determinant is

$|\det J|=\left|\begin{vmatrix}\mathbf x_u&\mathbf x_v\\\mathbf y_u&\mathbf y_v\end{vmatrix}\right|=\dfrac12$

So the integral is (NOTE: I'm guessing on what the integrand is supposed to be)

$\displaystyle\iint_R7xye^{x^2-y^2}\,\mathrm dA=\frac78\int_{u=0}^{u=10}\int_{v=0}^{v=4}e^{uv}(v^2-u^2)\,\mathrm dv\,\mathrm du$
$=\dfrac{117453-5733e^{40}}{3200}$