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

Benjamin Wright
7 min read
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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. Letβs get started!
#πͺοΈ 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.
#π₯ 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 (or approaching 0) and a radius of . As shown above, each cross-section is a super flat cylinder, with a volume of , where is the width and is the radius. Plugging in our formulas for width and radius, we get:
To add many of these volumes together, we use an integral! Our final integral is:
Where and are the stated boundaries or endpoints for the given equation , given as and .
#π₯ The Disc Method: Y-Axis
Sometimes, you will be asked to rotate an object around the ...

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