Volume with Disc Method: Revolving Around Other Axes

Abigail Young

7 min read

Next Topic - Volume with Washer Method: Revolving Around the x- or y-Axis

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Study Guide Overview

This study guide covers the disc method for finding volumes of solids of revolution, specifically focusing on rotations around axes other than the x and y-axes. It explains how to set up and solve integrals representing the volume, emphasizing the importance of visualizing the rotation and determining the correct variable of integration (x or y) based on the axis of rotation (horizontal or vertical). The guide provides the general integral formulas for both horizontal (y=b) and vertical (x=a) axes of rotation and includes a practice problem and solution involving rotating a region around a horizontal line.

#8.10 Volume with Disc Method: Revolving Around Other Axes

Welcome back to Fiveable AP Calculus! Today we are adapting the disc method to find volumes of objects that were rotated around an axis. If you want a quick refresher on the original disc method, click here for access to our 8.9 guide.

#🌪️Revolving Around Other Axes

Let’s get into it!

#⚾ The Basics

By “revolving around other axes” we mean that a function will revolve around either a horizontal (y=5) or vertical (x=3) axis. It is possible to revolve a function around other types of straight lines, but those transformations are more advanced than the scope of AP Calculus. So you might see…

!Untitled

Triangle revolved around a vertical line forming a cone

Courtesy of MathBits.com

Or you might see

!Untitled

Function revolved around the x-axis

Courtesy of Math24.net

But you won’t have to solve anything more complicated.

#🧠 How to Solve

Solving for the volume of a rotated shape with a different horizontal/vertical axis is actually pretty similar to solving for one with a standard x or y-axis! Remember that volume can be represented as an integral by adding up the area of many thin circular cross-sections, and don’t forget to draw out each problem to help you visualize what a rotation looks like.

#Step 1: Determine Your Method of Rotation

As a rule of thumb, if you rotate around a horizontal line (such as y=7 or the x-axis), you should write your integral in terms of x, which might look something like $\int_{a}...

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Previous Topic - Volume with Disc Method: Revolving Around the x- or y-Axis Next Topic - Volume with Washer Method: Revolving Around the x- or y-Axis

Question 1 of 9

🎉 If a region is revolved around the line $y = 4$ , which of the following integrals would be used to calculate the volume using the disc method?

$\int_{a}^{b} \pi (f(x) - 4)^2 dx$

$\int_{a}^{b} \pi (f(y) - 4)^2 dy$

$\int_{a}^{b} \pi (f(x))^2 dx$

$\int_{a}^{b} \pi (4 - f(x))^2 dx$