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## Integral Calculus

### Unit 3: Lesson 11

Volume: washer method (revolving around x- and y-axes)

# Generalizing the washer method

AP.CALC:
CHA‑5 (EU)
,
CHA‑5.C (LO)
,
CHA‑5.C.3 (EK)
Looking at the example from the last video in a more generalized way. Created by Sal Khan.

## Want to join the conversation?

• How would the volume be calculated if the shape was only partly rotated?
• This is a cool question!
If you know how far you want to rotate the shape (in radians) , you're area would be
A = ([angle of rotation]/2pi) * pi * ((f(x))^2-(g(x))^2)
You are essentially finding the area of a sector of a washer this way. Then you can proceed with your integral as usual.
• what happens if we evaluate the integral over interval [a,b] where on part of the interval f(x)>g(x) and on the other part g(x)>f(x)? i can visualize what the solid would look like but am not sure how the math would work out
• ^^Exactly, you would have to find their point of intersection and then split it there :)
• How would I find the volume of a shape like a bundt cake formed by a parabola with zeros at 1 and 5 being rotated around the y-axis
• When would you use the washer method as opposed to the disc method? Why is it hollow in the middle?
• The washer method should be used if there is "air" between the shaded region and the axis of rotation. When you rotate the shaded region, this air becomes a void in the shape.
• How do we know when to use the washer method or the disc method?
• You can always use either, the difference is that the washer method takes the cross-section of your final shape, then rotates it, while the disk method subtracts the entire volume of the shape enclosed by g(x) from the shape enclosed by f(x). If you think about it, both are the same thing, except in a slightly different order (using f(x)-g(x) at the end or the beginning).
(1 vote)
• So how exactly would you find the interval? Is it given to you or is it where the two functions intersect or something completely different? I wasn't too sure what he meant when discussing that portion.
• If two functions bound a region, then their intersection are usually the bounds. Finding the bounds can be tricky sometimes, especially in multivariable calculus - often my students have more difficulty with finding the bounds than doing the integration once the bounds have been identified. Why is that? It is usually due to being weak in algebra and not putting enough importance on graphing when it was being covered.
• how do you know when it's dy or dx and in which direction the washers are cut?
• When it's a dy integral, it's when you are rotating about a line that is vertical - for example, about x=2. (Sounds counterintuitive, dy for x=something, XD)

When it's a dx integral, it's rotated about a horizontal line, such as y = 5.
• I got to wondering why we are using for the squared radius f(x)^2 - g(x)^2 instead of [f(x) - g(x)]^2, and I played with it a little. Does using [f(x) - g(x)]^2 for the squared radius give you the volume if the axis of rotation is y=g(x)?