Question

Evaluate the given integral by making an appropriate change of variables, where r is the rectangle enclosed by the lines x - y = 0, x - y = 7, x + y = 0, and x + y = 6.

Answer 1

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


`\mathbf J=(\partial(u,v))/(\partial(x,y))=\begin{bmatrix}(\partial u)/(\partial x)&(\partial u)/(\partial y)\\\\(\partial v)/(\partial x)&(\partial v)/(\partial y)\end{bmatrix}=\begin{bmatrix}1&-1\\1&1\end{bmatrix}`

`\implies\det\mathbf J=2`

The area of the region is then given by


`\displaystyle\iint_R\mathrm dA=\int_(u=0)^(u=7)\int_(v=0)^(v=6)2\,\mathrm dv\,\mathrm du=84`