How Do You Calculate the Mass of a Cone Using Volume Integrals?

[implet]

Homework Statement

"A solid cone is bounded by the surface \theta=\alpha in spherical polar coordinates and the surface z=a. Its mass density is p_0\cos(\theta). By evaluating a volume integral find the mass of the cone.

Homework Equations

The Attempt at a Solution

I can't figure out the correct limits for the volume integral. Is it best to solve in Cartesian or spherical polar coordinates?

Many thanks :)

 

[Unco]

Homework Statement

"A solid cone is bounded by the surface \theta=\alpha in spherical polar coordinates and the surface z=a. Its mass density is p_0\cos(\theta). By evaluating a volume integral find the mass of the cone.

Your description of the cone suggests your interpretation of spherical polar coordinates is (r, \theta, \phi) where \theta is the angle from the positive z-axis and \phi is the angle from the positive x-axis.

We look to use these coordinates to calculate the integral for the cone. Sketch the cone: it makes an angle of alpha with the positive z-axis and goes up to z=a. More specifically...

\theta runs from 0 to \alpha.

\phi goes from ... to ... ?

To find the r-limits, draw a right-angled triangle:

   -----
   |   /
 a |  / r      where A is the angle alpha. 
   |A/
   |/

.