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## AP®︎/College Calculus BC

### Course: AP®︎/College Calculus BC>Unit 8

Lesson 9: Volume with disc method: revolving around x- or y-axis

# Disc method around x-axis

AP.CALC:
CHA‑5 (EU)
,
CHA‑5.C (LO)
,
CHA‑5.C.1 (EK)
Finding the solid of revolution (constructed by revolving around the x-axis) using the disc method. Created by Sal Khan.

## Want to join the conversation?

• How could he take pi out of the integral?
• ∫ c f(x) dx = c ∫ f(x) dx for any real number c, including pi.
• Why is the depth denoted by dx? Shouldn't it be delta x, like previous integral problems?
• delta x is a change in x.

dx means derivative x which is basically a super small change in x.
• Where's the +C? Is it unnecessary for solids? I would expect it at .
• +c is unnecessary for definite integrals because when you evaluate it at it's limits you will have one +c and one -c which will cancel out.
• I do not quite get the intuition of the shape when he rotates it, is there some other visualization ?
• Yes, there are many helpful computer programs by Peter Collingridge, that are quite helpful. Here is one of them:https://www.khanacademy.org/cs/3d-surface-viewer-2/1092842126 ,this one helped me understand. If this is not that good, see his other solids of revolution programs, as there are many. Yet, this is probably one of the most helpful as it was SPECIFICALLY made for this video.
• what are the units? were there any?
• On a coordinate plane, there are not any specific units, just x and y coordinates.
• Would y=x^2 the same as y=x^-2? on the interval from x=1 to x=3.
• No. y=x^-2 is the same as y=1/(x^2) because a negative exponent implies division.
• Is there a specific or technical name for these kind of problems? Like Volume Calculation of... I don't even know how to give an example...
• I think it is typically called a solid of revolution
• But isn't the edge of the disk curved?
• Yes, but we are finding the volume of an infinite number of infinitely thin cylinders. So thin that the curvature of the edge is negligible. Just like in Riemann sums where the square edge of the rectangle is so narrow that the error goes away.
• I understand this video well. But, what happens if the function that is being rotated about the x-axis has a negative portion? The function x^2 in this video did not go below the x-axis at any point, but a function such as f(x)=-x^4+2x^3-1 has a positive and a negative portion. Why do I have to make the limits of the integral the x-intercepts where f(x) is positive (i.e. x=1 to x=1.84)?