Formula for calculating the volume of a solid of revolution and illustrative examples

What is a solid of revolution? How to calculate the volume of a solid of revolution?

A solid of revolution is a shape created by rotating a plane around a fixed axis such as a cone of revolution, a cylinder of revolution, a sphere of revolution, etc. Below is the formula for calculating the volume of a solid of revolution, please refer to it.

Table of Contents

Formula for calculating the volume of a solid of revolution and illustrative examples
Common rotating solids: Cylindrical rotating solid, conical rotating solid, spherical rotating solid.

Calculate the volume of a circular block rotated around the Ox axis

If the circular block rotates around the Ox axis, the following formulas can be applied to calculate the volume of the rotating circular block:

Case 1 : Rotating circular block created by:

  • Line y= f(x)
  • x-axis y=0
  • x=a; x=b

Then, the formula for calculating volume is:

Case 2 : The rotating block is created by:

  • Line y= f(x)
  • Line y= g(x)
  • x=a; x=b

Then the formula for calculating the volume of a solid of revolution will be:

with

Calculate the volume of a circular block rotated around the Oy axis

If the circular block rotates around the Oy axis, the following formulas can be applied to calculate the volume of the rotating circular block:

Case 1 : The rotating block is created by:

  • Line x=g(y)
  • Vertical axis (x=0)
  • y=c; y=d

Then the formula for calculating the volume of a solid of revolution will be:

Case 2 : The rotating block is created by

  • Line x=f(y)
  • The equation x=g(y)
  • y=c; y=d

Then the volume of the solid of revolution will be:

with

Summary table of formulas for calculating the volume of a solid of revolution:

1. Vx generated by area S rotating around Ox:

Recipe :

Formula for calculating the volume of a solid of revolution and illustrative examples

2. Vx generated by area S rotating around Ox:

Recipe :

Formula for calculating the volume of a solid of revolution and illustrative examples

Example of calculating volume of solid of revolution

Example 1: 

Calculate the volume of the solid of revolution obtained by rotating the plane figure limited by the curve y = sinx, the x-axis and two straight lines x=0, x=π (drawing) around the Ox axis.

Formula for calculating the volume of a solid of revolution and illustrative examples

Solution

Applying the formula in the above theorem we have

Example 2: 

Calculate the volume of the solid of revolution obtained by rotating the plane figure bounded by the curve and the x-axis around the x-axis.

Formula for calculating the volume of a solid of revolution and illustrative examples

Prize:

We see:

For every x, this is therefore the equation of a semicircle with center O and radius R = A above the Ox axis. When rotated around the Ox axis, the plane will form a sphere with center O and radius R = A (figure). Therefore, we always have

 

So with this type of problem, we don't need to write the integration formula but can conclude based on the formula for calculating the volume of a sphere.

Example 3: 

Calculate the volume of the object lying between two planes x = 0 and x = 1, knowing that the cross-section of the object cut by plane (P) perpendicular to the Ox axis at the point with abscissa x(0≤x≤1) is a rectangle with two side lengths x and ln(x2+1).

Prize: 

Because the cross-section is rectangular, the cross-sectional area is:

We have the volume to calculate as

Example 4: Given a plane figure bounded by the lines y = 3x; y = x; x = 0; x = 1 rotating around the Ox axis. Calculate the volume of the resulting solid.

Formula for calculating the volume of a solid of revolution and illustrative examples

Prize:

The coordinates of the intersection of line x = 1 with y = x and y = 3x are points C(1;1) and B(3;1). The coordinates of the intersection of line y = 3x with y = x are O(0;0).

So the volume of the rotating solid to be calculated is:

Example 5 : Given a plane figure bounded by lines y = 2x2; y2 = 4x rotating around the Ox axis. Calculate the volume of the resulting circular solid.

Formula for calculating the volume of a solid of revolution and illustrative examples

Prize:

With is equivalent. The coordinates of the intersection of the line with are points O(0;0) and A(1;2).

So the volume of the rotating solid to be calculated is:

For problems requiring calculating the volume of a solid of revolution, you just need to use the correct formula for each case and pay attention when determining the boundary to be able to solve. Good luck.

Sign up and earn $1000 a day ⋙

Leave a Comment

Latest Grand Piece Online Code 04/25/2025 how to enter reward code

Latest Grand Piece Online Code 04/25/2025 how to enter reward code

Here are the Grand Piece Online Codes updated as soon as new codes are available. Including new codes and old codes that are still active.

8 Mistakes That Ruin Your TikTok FYP

8 Mistakes That Ruin Your TikTok FYP

Try to avoid making these mistakes while scrolling on TikTok to keep your FYP in line with your preferences.

How to use the FILTER function in Excel

How to use the FILTER function in Excel

If you regularly work with data sets in Excel, you know how important it is to be able to quickly find the information you need.

W3 Total Cache Plugin Vulnerability Exposes 1 Million WordPress Sites to Attacks

W3 Total Cache Plugin Vulnerability Exposes 1 Million WordPress Sites to Attacks

A critical bug in the W3 Total Cache plugin estimated to be installed on over a million WordPress websites has been discovered that could allow attackers to access a variety of information, including metadata on cloud-based applications.

Microsoft is about to stop deploying WSUS driver synchronization, users take note!

Microsoft is about to stop deploying WSUS driver synchronization, users take note!

In a new post on its Tech Community website, Microsoft has announced plans to discontinue WSUS driver synchronization in favor of its latest cloud-based driver services.

Samsungs GoodLock app gets a fresh new look

Samsungs GoodLock app gets a fresh new look

The GoodLock app on Samsung smartphones opens up a world of customization features.

How to cure insomnia for pregnant women in the last 3 months

How to cure insomnia for pregnant women in the last 3 months

The third trimester is often the most difficult time to sleep during pregnancy. Here are some ways to treat insomnia in the third trimester.

Phenomenon explained: strange circular halo around the Sun

Phenomenon explained: strange circular halo around the Sun

What is the phenomenon of the Sun being surrounded by a circle? This article will explain in detail to you the phenomenon of the circle around the Sun.

Apple: Siri Isnt Really Ready to Beat ChatGPT

Apple: Siri Isnt Really Ready to Beat ChatGPT

With the launch of the new generation iPad Mini, Apple has clearly affirmed that “AI-filled” software experiences are the way the company is heading in the future.

TikTok brings Getty Images to ads and AI-generated avatars

TikTok brings Getty Images to ads and AI-generated avatars

TikTok will allow advertisers to pull content from Getty Images when using the platform's AI ad creation tool.

AI Ads Will Appear More on TikTok Feeds

AI Ads Will Appear More on TikTok Feeds

No one likes to see ads, but this is one of the ways social media platforms can monetize the services they provide.

How to use the Chrome extension Trim to make Netflix better

How to use the Chrome extension Trim to make Netflix better

People used to spend a lot of time scrolling through Netflix, overwhelmed by decision fatigue and not knowing what to watch. But after finding the Chrome extension Trim, it completely changed their Netflix experience.

Latest Evade Roblox Codes and How to Redeem Codes

Latest Evade Roblox Codes and How to Redeem Codes

Evade game code allows you to receive important items for free, giving you an initial advantage in your survival journey.

How to play Taurus team DTCL season 14

How to play Taurus team DTCL season 14

Taurus TFT season 14 is a special clan with a permanent damage increase mechanism and the ability to drop gold when destroying enemies.

How to take beautiful night photos on smartphone

How to take beautiful night photos on smartphone

With the right techniques, you can take sharp night photos on your phone — sometimes even better than a professional camera.