Originally Posted by

**Spec** I guess I don't know how to solve any of the following, because I keep getting the wrong answer all the time. It's mostly the boundaries that I'm having trouble with.

1) $\displaystyle \int{\int_D{x^3}}dxdy$ where $\displaystyle D= \left\{(x,y)\in R^2; 1\leq x^2 + 9y^2 \leq 9; x \geq 3y\right\}$

[snip]

So here's what I tried to do:

1)

$\displaystyle \left\{\begin{array}{l}u= x\\v = \frac{y}{3}\end{array}\right.$

$\displaystyle |J(x,y)| = \frac{1}{3}$

Boundaries?

$\displaystyle 1 \leq u^2 + v^2 \leq 9$

$\displaystyle \left\{\begin{array}{c}u= rcos\varphi\\v = rsin\varphi\end{array}\right.$

$\displaystyle |J(r, \varphi)| = r$

Boundaries?

$\displaystyle \int{\int_D{r^3cos^3\varphi}}\cdot |J(u,v)| dxdy = \int{\int_D{\frac{r^4cos^3\varphi}{3}}}dxdy$

[snip]