I am having difficulties setting up the correct change of variable for the following two questions. I am wondering if someone can offer me assistance.
1) Let $\ n > 0$, Compute the integral
$$\iint_\Omega \ (x+y)^n \,dx\,dy$$
where $\Omega=\{(x,y): x\geq 0,\ y\geq 0, x+y \leq 1\}$
2) Evaluate the integral
$$\int_{0}^{1} \int_{0}^{x} \sqrt{{x}^{2}+{y}^{2}} \,dy\,dx$$
using the transformation $\ x=u, y=uv\ $
For the first question, I tried
$$\frac{1} {2} \int_{0}^{1} \int_{-v}^{v} {v}^{n}\ du\ dv,\ $$ with $\ u=x-y,\ v=x+y\ $ but I don't seem to be getting the correct answer. The textbook solution is $\frac{1} {6(n+4)}\ $
For the second question, I figured the original is a triangular region with an area $\frac{1} {2}\ $, so when applying a change of variable, the transformed region should also have the same area. I tried with the following transformation:
$$\int_{0}^{1} \int_{\frac{1} {v}}^{1} \ v\sqrt{{u}^{2}+{(uv)}^{2}}\ du\ dv\ $$
However, I got the wrong answer. The textbook solution is $\frac{1} {6}\ [\sqrt{2}+log(1+\sqrt{2})] \ $
Thank you in advance
For number 1, I don't get the answer in the book. I have $$ \int_{0}^{1} \int_{0}^{1-y} {(x+y)^n}\ dx\ dy=\int_{0}^{1} \int_{u}^{1} {v^n\ du\ dv}$$ using the transformation, $u=x, v=x+y$ which has Jacobian determinant $1$. Evaluating the integral gives $\frac{1}{n+2}.$ This is the same answer I get from Wolfram Alpha.