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

• Calculate the volume of a circular block rotated around the Ox axis
• Calculate the volume of a circular block rotated around the Oy axis
• Example of calculating volume of solid of revolution

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 :

2. Vx generated by area S rotating around Ox:

Recipe :

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.

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.

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.

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.

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.