Jacobian of the Transformation Problem, Multivariable Calculus

Question

I have the following Jacobian problem:

I'm having trouble working through it because the double integral in terms of u and v is throwing me off. Could someone walk me through it? Thanks!

Answer 1

Start with setting up the double integral and then subbing in the new $x,y$ values! Notice $0 \leq v \leq 3$ and $v \leq u \leq v+2$ (these are the functions bounding $u$) so we get: $$\int_{0}^{3} \int_{v}^{v+2} {u^2v-2uv^2+v^3dudv}$$ Now subbing in the $x$ and $y$ values it works out nicely (don't forget to change the endpoints):$$\int_{0}^{3} \int_{y-y}^{y-y+2} {(x+y)^2y-2(x+y)y^2+y^3dxdy}={\int_{0}^{3} \int_{0}^{2}} {x^2y+y^3+2xy^2-2xy^2-2y^3+y^3dxdy}={\int_{0}^{3} \int_{0}^{2}} {x^2y \hspace{2mm} dxdy}={\int_{0}^{3}} {\frac83y \hspace{2mm} dy}=12$$