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erik [133]
3 years ago
11

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.
Mathematics
1 answer:
vova2212 [387]3 years ago
6 0
\begin{cases}u=x-y\\v=x+y\end{cases}

\mathbf J=\dfrac{\partial(u,v)}{\partial(x,y)}=\begin{bmatrix}\dfrac{\partial u}{\partial x}&\dfrac{\partial u}{\partial y}\\\\\dfrac{\partial v}{\partial x}&\dfrac{\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
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