$(1$ point$)$

Consider the transformation $T$:$x=\frac{40}{41}u-\frac{9}{41}v,y=\frac{9}{41}u+\frac{40}{41}v$

A$.$ Compute the Jacobian:

$\frac{\text{d}\text{e}\text{l}(x,y)}{\text{d}\text{e}\text{l}(u,v)}=-\frac{1519}{1681}$

B$.$ The transformation is linear, which implies that it transforms lines into lines. Thus, it transforms the square $S$:$-41u41,-41v41$ into a square $T(S)$ with vertices:

C$.$ Use the transformation $T$ to evaluate the integral ${}_{T(S)}{x}^2+y^{2}dA$