Question

Use the given transformation to evaluate the integral.

∬R (x - 8y) dA, where R is the triangular region with vertices (0, 0), (7, 1), and (1, 7). x = 7u + v, y = u + 7v

Answer 1

For the transformation

$\begin{cases}x=7u+v\\y=u+7v\end{cases}$

the Jacobian is

$\dfrac{\partial(x,y)}{\partial(u,v)}=\begin{bmatrix}7&1\\1&7\end{bmatrix}$

with determinant

$\det\left(\begin{bmatrix}7&1\\1&7\end{bmatrix}\right)=48$

The vertices of the triangle in the $u,v$-plane are

$(x,y)=(0,0)\implies(u,v)=(0,0)$

$(x,y)=(7,1)\implies(u,v)=(1,0)$

$(x,y)=(1,7)\implies(u,v)=(0,1)$

Then the integral is

$\displaystyle\iint_R(x-8y)\,\mathrm dA=-48\int_0^1\int_0^{1-v}(u+55v)\,\mathrm du\,\mathrm dv=\boxed{-448}$