Use the change of variables $x=\sqrt{u+v}$, $y=v$ to evaluate: $\iint x\,dx\,dy$
over the region R in the first quadrant of the $xy$-plane bounded by $y=0$, $y=16$, $y=x^2$, and $y=x^2−9.$
I'm pretty lost. I tried to find the Jacobian, and I got $(1/2\sqrt{u+v})-(1/2\sqrt{v+u})$. I also tried to find the bounds for $x$ and $y$. However, I have no idea if I'm even on the right track.
$x = \sqrt {u+v}\\y = v$
jacobian:
$dy\ dx = $$\det \pmatrix {\frac {\partial x}{\partial u}&\frac {\partial x}{\partial v}\\\frac {\partial y}{\partial u}&\frac {\partial y}{\partial v}}\ du\ dv\\ \det \pmatrix{\frac {1}{2\sqrt {u+v}}&\frac {1}{2\sqrt {u+v}}\\ 0&1} = \frac {1}{2\sqrt {u+v}} \ du\ dv$
Limits:
$y = x^2\\ v = u+v\\ u = 0$
$y = x^2-9\\ u = 9$
The limits for v:
$y = 0\\ v = 0$
$y = 16\\ v = 16$
$\int_0^{16}\int_0^{9} \frac 12 \ du\ dv$