Change of Variables in Multiple Integrals

Quote:

Originally Posted by angel.white

Having a really tough time figuring out how to set this problem up:

Use the given transformation to evaluate the integral.

$\int \int_R x^2 ~dA$ , where R is the region bounded by the ellipse $9x^2+4y^2 = 36$ ;

$x=2u$

$y=3v$

$dA = dx \, dy \rightarrow J(u, v) \, du \, dv = [(2)(3) - (0)(0)] \, du \, dv = 6 \, du \, dv$ .

$R_{xy}: ~ 9 x^2 + 4 y^2 \leq 36 ~ \rightarrow R_{uv}: ~ 36 u^2 + 36 v^2 \leq 36 \Rightarrow u^2 + v^2 \leq 1$ .

So the double integral becomes $\int \int_{R_{uv}} (2u)^2 (6) \, du \, dv = 24 \int \int_{R_{uv}} u^2 \, du \, dv$ .

To exploit the circular region in the uv-plane I'd suggest switching to polar coordinates now to seal the deal.