finding cylinder mass using triple integrals

Question

Hi, I'm trying to find mass of a cylinder using triple integrals.

The data I have:

- height = h
- base is a circle and its radius = a
- the density in a point P is directly propotional to the distance of its basis.

I think that I can use p(z) = k*z

and that the basis formula is

x^2 + y^2 = a^2 (the circunference formula).

I also think that the inferior limit in z integral is 0 and the superior limit is h.

But I can't figure out the x and y inferior and superior limits.

Someone can help me?

Answer 1

The triple integral is 

 

M = ∫dx ∫dy ∫dz  k z

Since the boundary condition on x and y is the circle of radius a,   the easiest way to proceed is to change

 ∫ dx ∫ dy  to  ∫ρ dρ ∫ dθ  That is - change to polar coordinates.    The integral over  θ is from 0 to 2pi and the integral over  ρ is from 0 to a.    Since the integrand does not depend on x or y, it does not depend on ρ or θ.    Thus the integral over ρ and evaluates to  pi a² .   This leaves

 

   M  =  pi a² ∫ kz dz.     Since the limits on this integral are 0 and h, 

 

   M =   pi a² (1/2)k h²  

 

    Since the units of k are   kg/m⁴    , the units of M are kg as should be the case.

 

   The factor  pi a²    is just the area of the base.