Use cylindrical coordinates to evaluate the triple integral
∫∫∫sqrt(x2+y2)dV where E is the solid bounded by the circular parabaloid
z=9−16(x2+y2) and the xy plane.

Question

Use cylindrical coordinates to evaluate the triple integral
∫∫∫sqrt(x2+y2)dV where E is the solid bounded by the circular parabaloid
z=9−16(x2+y2) and the xy plane.

anonymous · Answer

This was the integral that I set up:

$$\int\limits_{0}^{2\pi}\int\limits_{0}^{3/4}\int\limits_{0}^{9-16r^2}rdzdrd \theta$$

And I was not able to wind up with the correct answer.

anonymous · Answer

you do z first

anonymous · Answer

you integrate and get z b/c u only have a dz

anonymous · Answer

so u get 9-16r^{2} rdrdtheta

anonymous · Answer

I know I just think that my second limit of integration might be wrong. I know how to evaluate the integral.

anonymous · Answer

then for r you get 9r-16r^3 drdtheta

anonymous · Answer

$$\int\limits_{0}^{3/4} 9r-16r^{3} drd$$

anonymous · Answer

$$\frac{ 9r^2 }{ 2} - 4r^4  $$

anonymous · Answer

from 0 to 3/4

anonymous · Answer

no there is a function r, so it should be r^2.
then int[9r2-16r4]0-3/4  *2pi

anonymous · Answer

r3-16/5r5]0-3/4 * 2pi is the final answer.