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/
   |/

.