Thread: Double Integral Using Polar Coordinates to Find Mass of Disk

Problem: A disk of radius 5 cm has density 15 gm/cm62 at its center, density 0 at its edge. Assume its density is a linear function of the distance from the center.

Set up the integral to find the total mass of the disk.

My Attempt So Far:

I understand how to set up the integral itself in polar coordinates just fine.

The integral I set up is as follows:



The part I'm actually having trouble with is determining what the function I'm integrating is!

In the problem, it says that the density of this disk is 15gm/cm^2 at the center, and 0gm/cm^2 at the edge. It says that the function is a linear function of the distance from the center.

This is what I drew based on that, with z sort of representing the density.



This graph is in the xy plane (I forgot to label it). I'm not really sure where to go from here, or if this is the right way to look at it. How do I get an equation representing the change in density of the disk?

Edit:

I'm still playing with this and I've set up the following idea:

I know that when r = 0, density = 15, and when r = 5, density= 0, which gives me two points (0,15) and (5,0). If I make these into a linear function, I get -3r. Is that anywhere close to right?

No, I'd say you need to try again with your density function. The density-intercept of -3r is not the required 15. I think you have the right slope, though. If the density is f(r) = -3r + b, find b. Plug that into your integral.

Thank you so much! I can't believe I didn't see that before...

You're welcome. Have a good one!