If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

AP®︎ Calculus AB (2017 edition)

Course: AP®︎ Calculus AB (2017 edition)>Unit 11

Lesson 8: Volume: shell method (optional)

Calculating integral with shell method

Evaluating integral set up with shell method for two functions. Created by Sal Khan.

Want to join the conversation?

• This is the same question he answered using the washer method. Sal arrive to the same answer as the washer method. So the method he used can be use to get the same answer as long as the same question does not change?
• He is trying to show with a worked example that the shell method really does work (granted that you believe in the disc method). That is the great thing about math: there are so many different ways to arrive at an answer, and they all work!
• How and when we use the shell method? I m confused how can I classify it to other methods like Disc and Washer.
• In BASIC questions, they will give you y = f(x) as the main "funny boundary" (maybe y = x² or y = sin(x) ), while the other sides of the region will be horizontal or vertical lines (x-axis, y-axis, x = 3, etc.). Or possibly y1 = f1(x), y2 = f2(x) for the "top" and the "bottom" of the region. In these cases, here is the idea:
1) IF the region is then rotated around a horizontal line (x-axis, or y = k), then you probably want to use discs or washers (depending on whether there is a hole in the middle). This is because slicing the shape with a straight vertical knife will give discs or washers, and the radius is determined by the "curvy" function y = f(x).
2) IF the region is rotated around a vertical line (y-axis, or x = k), then you probably want to use cylindrical shells. This is because slicing the shape into shells will give you shells whose height is determined by the "curvy" function y = f(x).
In both of these cases, you would end up doing a "dx" integral.

As always, there can be tricky exceptions to this general rule. For example, if they give you x = f(y), then everything is reversed, so that you end up with a "dy" integral: discs/washers if rotated around a vertical line and shells if rotated around a horizontal line.
• Why aren't the limits for the volume 0 and 2?
• The bounds are the places were the two functions intersect.
• why is the integral in terms of x? can it be in terms of y?
• The integral has to be taken in terms of what axis the shells or washers are stacked about.
• I had a general question on Solids of Revolution. When/How do we know when we are to use dy or dx and when/how do we get to know when to use the shell or the disk/washer method or does it not matter ?
(1 vote)
• I'm not aware of any general answer to this question. Many problems can be solved either way with about equal difficulty. Depending on the formulas involved, in some cases you may find that one way forces you to solve a difficult integral while the other way gives you an easy one, and it may not be obvious which way is easier until you work on it a while.
• Why can't you do surface area of the the top the the shell times the height. Ie. Hollowing out a cylinder?
• why is the constant not added while taking the integral of the expressions ??
(1 vote)
• When we have a definite integral, we don't add a constant. We are just using the first fundamental theorem of calculus:
b
∫ f(x) dx = F(b) - F(a)
a
However, if we have an indefinite integral, we have to write +C at the end.