How Do You Calculate the Total Mass of a Cylinder with Variable Density?

[TheSpaceGuy]

Total mass?

Homework Statement

Find the total mass of the part of the solid cylinder x^2 + y^2 ≤ 4 such that x^2 ≤ z ≤ 9 - x^2 , assuming that the mass density is p(x, y, z) = I y I (absolute value of y).


I have heard about center of mass but this is throwing me off?

The Attempt at a Solution

Thats where the problem is.

 

[jbunniii]

All you have to do is integrate p(x,y,z) over the volume of interest.

 

[TheSpaceGuy]

But how would I get the limits of integration. How about choosing x from 0 to 4 and y is x^2 to 4? z is given. Am I on the right track?

 

[HallsofIvy]

Did you think about this very long? x can't 4 and satisfy $x^2+ y^2= 4$ for any y! And I have no idea how you got "y is x^2 to 4"! What do you get if you solve $x^2+ y^2= 4$ for y?

Perpendicular to the z-axis, the boundary is the cylinder $x^2+ y^2= 4$. You could let x very from -2 to 2 and, then, for every x, y varies from $-\sqrt{4- x^2}$ to $\sqrt{4- x^2}$. Or write it in polar coordinates with r going from 0 to 2, $\theta$ from 0 to $2\pi$.

For every point (x, y), the z-coordinate varies from $x^2$ to $9- x^2$ just as you are told.