Using integrals to get volume, center of mass, and surface area

[ns5032]

Homework Statement

For the homogeneous ice-cream cone that is given in spherical coordinates by rho= pi/4 (the bottom part) and phi=cos(rho) (the top part), find the volume, the center of mass, and the surface area. ((You have to do this problem using integrals, known formulas from elementary geometry can be helpful, but they are not sufficient for justifying the answer)).

Homework Equations

According to spherical coordinates, "rho" is distance from the origin and "phi" is the angle between the z-axis and the line connecting the origin and the point.

The Attempt at a Solution

I know that to find the volume, I must do a triple integral. The outside integral's limits will be in theta (the angle between the line from the origin to the point and the x-axis), the middle integral's limits in terms of phi, and the inside in terms of rho. I do not know what to put inside the triple integral.

For the center of mass, I must use the double integral of the density function, and then I think I can get it from there, but I am not sure how to get the density function.

To find the surface area, I believe I must do the double integral of the magnitude of vector rrho X rphi. How do I get these vectors??

Need major help! Thanks!

 

[Office_Shredder]

To find the volume, you just integrate over 1dxdydz. Then you replace dxdydz with your spherical coordinates' Jacobian to have much nicer limits of integration

 

[ns5032]

How do I go about switching the limits to spherical coordinates