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 [itex]x^2+ y^2= 4[/itex] for any y! And I have no idea how you got "y is x^2 to 4"! What do you get if you solve [itex]x^2+ y^2= 4[/itex] for y?

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

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