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  1. AP Calculus
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Volume with Disc Method: Revolving Around the x- or y-Axis

Benjamin Wright

Benjamin Wright

7 min read

Next Topic - Volume with Disc Method: Revolving Around Other Axes

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Study Guide Overview

This study guide covers calculating the volume of solids of revolution using the disc method. It explains how to revolve shapes around the x-axis and y-axis, set up the definite integrals, and evaluate them. Practice questions demonstrate applying these steps for rotations around both axes. The guide focuses on single functions and revolving around x or y-axis only.

#8.9 Volume with Disc Method: Revolving Around the x- or y-Axis

In AP Calculus, we dive deep into the concept of calculating volume through the disc method, where we revolve shapes around either the x- or y-axis. This method is fundamental in solving problems involving volumes of solids of revolution. Let’s get started!

#πŸŒͺ️ Volumes of Solids of Revolution

When we talk about finding the volume of a solid of revolution, we're essentially determining how much space the 3D shape occupies. To do this, we take a curve and rotate it around a particular axis, which creates a solid shape. The disc method allows us to calculate the volume of these solids accurately.

#πŸ₯ The Disc Method: X-Axis

The disc method involves slicing the solid into infinitely thin discs perpendicular to the axis of rotation. By summing up the volumes of these discs using definite integrals, we obtain the total volume of the solid.

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A Rotated Solid with a cut-out cross section that is disk-shaped with a width of dx

Courtesy of Cochran Math

To find the volume of a solid rotated around the x-axis, as shown above, we add together the volume of many thin cross-sections! These cross sections of the rotate volume have a width of dxdxdx (or approaching 0) and a radius of f(x))f(x))f(x)). As shown above, each cross-section is a super flat cylinder, with a volume of wΓ—Ο€r2w \times \pi r^2wΓ—Ο€r2, where www is the width and rrr is the radius. Plugging in our formulas for width and radius, we get:

Ο€(f(x))2dx\pi (f(x))^2 dxΟ€(f(x))2dx

To add many of these volumes together, we use an integral! Our final integral is:

∫abΟ€(f(x))2dx\int_{a}^{b}\pi (f(x))^2dx∫ab​π(f(x))2dx

Where aaa and bbb are the stated boundaries or endpoints for the given equation f(x)f(x)f(x), given as x=ax = ax=a and x=bx=bx=b.

#πŸ₯ The Disc Method: Y-Axis

Sometimes, you will be asked to rotate an object around the ...

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Previous Topic - Volumes with Cross Sections: Triangles and SemicirclesNext Topic - Volume with Disc Method: Revolving Around Other Axes

Question 1 of 10

πŸŽ‰ What best describes a solid of revolution?

A 2D shape

A 3D shape made by rotating a curve around an axis

A flat disk

A random 3D shape