Let $R$ be the square with vertices $(0,0), (1,-1), (2,0), (1,1)$. Evaluate $\displaystyle \iint_R\sqrt {\frac{x-y}{x+y+1}}$.
I realized one way to approach this problem is to divide it into 4 equal pieces of the same size and perform integration on 1 of the piece, multiplied by 4. Using the top left piece, $0\leq x \leq 1, 0\leq y\leq x$.
Please drop some hints on how to evaluate $\displaystyle \iint_R\sqrt {\frac{x-y}{x+y+1}}$.
Best Answer
The region of integration and the form of the integrand is practically begging to be transformed via a rotation by $\pi/4$ radians. Try $$u = x-y, \quad v = x+y,$$ which then turns the region of integration into $$(u,v) \in [0,2] \times [0,2].$$ Then compute the Jacobian of this transformation, which is $2$, and you should get $$\int_{u = 0}^2 \int_{v = 0}^2 2 \sqrt{\frac{u}{v+1}} \, dv \, du,$$ which is much easier to evaluate.