Question

Find the volume of the region in the first octant that lies between the cylinders r=1 and r=2 (i.e. the cylinders centered along the z-axis with radius 1 and 2) and that is bounded below by the xy-plane and above by the surface z=xy.

Answer 1

I think all you have to do is convert xy to cylindrical coordinates, which leaves the bounds as\[0\le \theta \le2\pi\]\[1\le r \le2\]\[0\le z \le r^2\sin \theta \cos \theta\]the integral is then\[\int\limits_{0}^{2\pi}\int\limits_{1}^{2}\int\limits_{0}^{r^2\sin \theta \cos \theta}rdzdrd \theta\]

Answer 2

Then just solve for the integral right??

Answer 3

yes :)

Answer 4

volume element in cylindrical coordinates:\[dV=rdrdzd \theta\]integrate both sides within the proper bounds to get the formula above.

Answer 5

r is the jacobian right?

Answer 6

i got the volume equal to zero, help??