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136 LearnersLast updated on December 11, 2025
Volume of Solid of Revolution
The volume of a solid of revolution is the total space it occupies when a two-dimensional shape is rotated around an axis. This concept is used in calculus to find the volume of complex shapes by integration. In real life, this can be related to finding the volume of objects like vases, bowls, or any objects with rotational symmetry. In this topic, letโs learn about the volume of solids of revolution.
What is the volume of a solid of revolution?
The volume of a solid of revolution is the amount of space it occupies. It is calculated using the method of integration.
One common method is the disk method, where you integrate the area of circular disks along the axis of rotation. Another is the shell method, which uses cylindrical shells.
How to Derive the Volume of a Solid of Revolution?
To derive the volume of a solid of revolution, we use calculus, specifically integration. The disk method involves slicing the solid perpendicular to the axis of rotation into thin disks and summing their volumes.
The formula for the volume using the disk method is: \(V = \pi \int_{a}^{b} [f(x)]^2 \, dx ]\)where f(x) is the function being rotated around the axis.
How to find the volume of a solid of revolution?
The volume of a solid of revolution can be found using integral calculus. First, determine the axis of rotation and the function or region being rotated. Use the disk or shell method to set up an integral that represents the volume.
For the disk method, the formula is: \( V = \pi \int_{a}^{b} [f(x)]^2 \, dx \)
For the shell method, the formula is:\( V = 2\pi \int_{a}^{b} x \cdot f(x) \, dx
\)
Evaluate the integral to find the volume.
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Tips and Tricks for Calculating the Volume of Solids of Revolution
Choose the appropriate method: Decide between the disk and shell method based on the axis of rotation and the function given.
Understand the geometry: Visualize the shape formed by the revolution, which helps in setting up the integral correctly.
Simplify calculations: If possible, simplify the function before integrating to make the calculations easier.
Check units: Make sure your final answer is in cubic units, as volume is a measure of space.
Common Mistakes and How to Avoid Them in Volume of Solids of Revolution
Making mistakes while learning the volume of solids of revolution is common. Let’s look at some common mistakes and how to avoid them to get a better understanding of these volumes.
Volume of Solid of Revolution Examples
Problem 1
Find the volume of the solid obtained by rotating the region bounded by \( y = x^2 \) and \( y = 0 \) from \( x = 0 \) to \( x = 1 \) about the x-axis.
The volume of the solid is \((\frac{\pi}{5}).\)
Explanation
Using the disk method:\( V = \pi \int_{0}^{1} (x^2)^2 \, dx \)
\(V = \pi \int_{0}^{1} x^4 \, dx \)
\(V = \pi \left[ \frac{x^5}{5} \right]_{0}^{1} \)
\(V = \frac{\pi}{5} \)
Problem 2
Calculate the volume of the solid formed by rotating the function \( y = \sqrt{x} \) from \( x = 0 \) to \( x = 4 \) about the x-axis.
The volume of the solid is \((\frac{32\pi}{3}).\)
Explanation
Using the disk method:\( [ V = \pi \int_{0}^{4} (\sqrt{x})^2 \, dx ]\)
\([ V = \pi \int_{0}^{4} x \, dx ]\)
\([ V = \pi \left[ \frac{x^2}{2} \right]_{0}^{4} ]\)
\[ V = \pi \left( \frac{16}{2} \right) \]
\([ V = 8\pi ]\)
Problem 3
Determine the volume of the solid obtained by rotating the region between \( y = x + 1 \) and \( y = 0 \) from \( x = 0 \) to \( x = 2 \) about the y-axis.
The volume of the solid is\( (8\pi).\)
Explanation
Using the shell method:\( [ V = 2\pi \int_{0}^{2} x(x+1) \, dx ]\)
\([ V = 2\pi \int_{0}^{2} (x^2 + x) \, dx ]\)
\([ V = 2\pi \left[ \frac{x^3}{3} + \frac{x^2}{2} \right]_{0}^{2} ]\)
\([ V = 2\pi \left( \frac{8}{3} + 2 \right) ]\)
\([ V = 8\pi ]\)
FAQs on Volume of Solid of Revolution
1.Is the disk method always better than the shell method?
2.How do you choose between the disk and shell methods?
3.What if the axis of rotation is not the x or y-axis?
4.Can the function being rotated be a piecewise function?
5.What units are used for the volume of a solid of revolution?
Important Glossaries for Volume of Solid of Revolution
- Disk Method: A technique to find the volume of a solid of revolution using disks perpendicular to the axis of rotation.
- Shell Method: A technique to find the volume using cylindrical shells parallel to the axis of rotation.
- Integration: A mathematical process to sum areas or volumes to find total quantities, such as volume.
- Axis of Rotation: The line around which a shape is rotated to create a solid of revolution.
- Cubic Units: Units used to measure volume, such as cubic centimeters (cm³) or cubic meters (m³).
Seyed Ali Fathima S
About the Author
Seyed Ali Fathima S a math expert with nearly 5 years of experience as a math teacher. From an engineer to a math teacher, shows her passion for math and teaching. She is a calculator queen, who loves tables and she turns tables to puzzles and songs.
Fun Fact
: She has songs for each table which helps her to remember the tables
