so I need to solve the following integral using some change of variables:
$$\underset{[0,{\pi}]\times[0,{\pi}]}{\iint}{\cos}{(x+y)}dxdy$$
It's easy without a change of variables, of course, but any suggestion as to how to do it with a change of variables?
Thanks
Use the substitution $$\left.\eqalign{x&=u \cr y&=v-u\cr} \ \right\}\qquad\bigl((u,v)\in P: \ 0\leq u\leq\pi , \ \ u\leq v\leq u+\pi\bigr)\ .$$ Its Jacobian determinant is $\equiv1$. Therefore you get $$\eqalign{\int_{[0,\pi]\times[0,\pi]}\cos(x+y)\>{\rm d}(x,y)&=\int_P\cos v\>{\rm d}(u,v)=\int_0^\pi\int_u^{u+\pi}\cos v\>dv\>du\cr &=\int_0^\pi-2\sin u\>du=-4\ .\cr}$$