Evaluate Double Integration Evaluate $\iint−3x^2 dA$ over the region in the first quadrant bounded by the hyperbola xy=16 and the lines $y=x$, $y=0$, and $x=8$.
I have drawn a picture, but I am still a little unsure on what to bound my integrals from. I have from $0$ to $8$ for $dx$ and $0$ to $4$ for my $dy$, but I get a negative answer when I compute my answer. 

Outline: Making a good sketch is the key step. Note that the line $y=x$ meets the hyperbola at $(4,4)$.
It is convenient to break up the region of integration into two parts, the part with $0\le x\le 4$, and the part with $4\le x\le 8$.
For the integral over the first part, express as an iterated integral. Integrate first with respect to $y$, where $y$ goes from $0$ to $x$, and then with respect to $x$, where $x$ goes from $0$ to $4$.  
For the integral over the second part, $y$ goes from $0$ to $\frac{16}{x}$, and then $x$ goes from $4$ to $8$.
Remark: If the integrand really is $-3x^2$, as it appears to be, then the integral should be negative.
