Question

Use the given transformation to evaluate the integral. (x ? 10y) dA, R where R is the triangular region with vertices (0, 0), (9, 1), and (1, 9). x = 9u + v, y = u + 9v

Answer 1

The given change of coordinates has Jacobian

$\mathbf J=\begin{bmatrix}\dfrac{\partial x}{\partial u}&\dfrac{\partial x}{\partial v}\\\\\dfrac{\partial y}{\partial u}&\dfrac{\partial y}{\partial v}\end{bmatrix}=\begin{bmatrix}9&1\\1&9\end{bmatrix}$

so the area element is

$\mathrm dA=\mathrm dx\,\mathrm dy=|\det\mathbf J|\,\mathrm du\,\mathrm dv=80\,\mathrm du\,\mathrm dv$

The new region is a right triangle with vertices (0, 0), (1, 0), and (0, 1) in the $u,v$ plane.

Then the integral becomes

$\displaystyle\iint_R(x-10y)\,\mathrm dA=-80\int_{u=0}^{u=1}\int_{v=0}^{v=1-u}(u+89v)\,\mathrm dv\,\mathrm du=-1200$