# Thread: Limits of Integration for Shell Method

1. ## Limits of Integration for Shell Method

The Shell method makes total sense, except for finding limits of integration.

How do you find them if rotating on the y axis (when you use the form y = "x variables here") or when rotating on the x axis (when you use the form x = "y variables here")

Here are two problems. How to find limits of integration?

1. region bounded by the curve $y = \sqrt{x}$ and the x axis, and the line x = 4 revolved around the y axis

2. region bounded by the curve $y = \sqrt{x}$, the x axis, and the line x = 4, revolved about the x axis.

2. ## Re: Limits of Integration for Shell Method

I think I got somewhat of an answer for this, but need to confirm.

Ok, In the case of the shell method, and being revolved around the y axis, then plug in the given x value for the line (x = 4) into the given y equation. That gives you one limit of integration, you can assume the other limit of integration is 0. So in the 1st problem the limits of integration are $y = \sqrt{4} = 2$

In the case of the shell method, revolving around the x axis, you would set "x = something" equal to the line equation of "x = 4", in other words, something = 4, to find the y value. In the case of problem two. You would have

$y = \sqrt{x}$

$y^{2} = 4$

$y = 2$

So you can say the limits of integration are 2 and 0 (the other limit of integration is assumed to be 0)

3. ## Re: Limits of Integration for Shell Method

TLDR, but the way I proved to myself that what I was doing was correct was by finding the volume of the solid of revolution using both the shell(disk) and washer(cylinders) method, and then seeing that both gave me the same value. So I knew that one of them was wrong if they both didn't have the same value and also I knew for certain that I did it right when both methods yielded the same answers, and eventually I really understood it well.

4. ## Re: Limits of Integration for Shell Method

The method of thinking which I posted in the top two posts, seems to work for any shell integration problem (or maybe even surface area problems). Of course sometimes though, the bottom line isn't the x axis (y = 0), in that case, it's a little different. Usually those aren't on exams though. Of course, generally speaking, the bottom line isn't used in calculating limits of integration, nor in solving the problem.