How to Evaluate a Triple Integral Bounded by a Cylinder and Planes?

[Pete_01]

Homework Statement

Evaluate the triple integral where E is the solid bounded by the cylinder y^2+ z^2 = 576 and the planes x = 0, y = 4 x and z =0 in the first octant.

Homework Equations

The Attempt at a Solution

I figure that by solving for z I can get the bounds, so between 0 to sqrt(576-y^2) would be the z upper and lower bounds. X bounds would be between 0 and 24? And I am not sure about y.

 

[HallsofIvy]

I presume you mean "the triple integral for the volume"- i.e. just integrate dV= dxdydz over that region. Otherwise you haven't said what it is you want to integrate! I think, because of the spherical symmetry here, using spherical coordinates would be best. Obviously $\rho$ will go from 0 to $\sqrt{576}= 24$, $\phi$ from 0 to $\pi/2$ and, since y= 4x has slope 4, $\theta$ will go from 0 to arctan(4).