# Thread: Volume of a region

1. ## Volume of a region

Referring to the figure above, find the volume generated by rotating the region about the line OC.

So I'm trying to find the volume of R1, but don't know where to start. if I solve for x in terms of y (for y=x^3), wouldn't that just give me a completely new volume?

2. Have you learned the techniques for calculating the volume of regions that are rotated around an axis? That is what this problem is asking you to do. In general there is the "shell" method and the "washer" or "disc" method.

Does this ring any bells?

3. I don't know the shell method, but I do know of the washer/disc. Since I've just begun I'm more familiar with doing this with the x-axis rather than the y-axis. But from what I know, I'm supposed to solve for x in terms of y to rotate it around the y-axis. Then, I use the area of a known shape, a circle in this case, and plug in the radius and integrate it.

4. Yes, that's all correct. I guess your confusion is with solving for x. This gives you $\displaystyle x=\sqrt[3]{y}$. Don't be concerned, this is actually the same exact curve as before, it's just solved for a different variable! So the volume you will get is still correct.

So, you can set up your integral like so:

$\displaystyle \pi \int_0^1 (R^2 - r^2) dy$

where $\displaystyle R$ is the outside radius of the "washer" or "disc" and $\displaystyle r$ is the inside radius.

The only difference is we are now integrating on y instead of x, and the two radiuses are in terms of y instead of x.

5. ah alright, I get it now. thanks!