Exam Questions - Volume of revolution about the x-axis

Revision Notes: Volume of Revolution About the X-axis

The Volume of Revolution is the volume of a three-dimensional figure that is created when a two-dimensional shape or curve is rotated around the x-axis.
This concept is commonly tested in Further Maths papers, particularly involving graphs, shapes and functions.
Understanding the basic theory of calculus is crucial to solve these problems.

For a curve given by a function y=f(x) between the x-values a and b, the volume V of revolution around the x-axis is given by the formula:

V = π ∫[a to b] (f(x))^2 dx
You obtain the volume by integrating (f(x))^2 from a to b, and then multiplying the result by pi (π).

Identify the curve given and the limits of rotation (values a and b) about the x-axis in the question.
Sketch the curve if a graph is not already provided. This can give a better understanding of the shape being rotated.
Put the equation of the curve y=f(x) into the volume formula and carry out the integration.
Be careful with negative areas. The ‘volume’ is always a positive value so ignore any negative signs.
Remember to include the π in the formula. It’s a common mistake to forget this important constant.
Double check your calculations and make sure your final answer makes sense in the context of the question.

These questions require a good understanding of calculus including forming an integral and evaluating it.
Make sure that you feel confident in your calculus skills including integration techniques such as integrating power functions, fractions and trigonometric functions as these are essential for calculating the Volume of Revolution.

One common mistake is to not square the function in the integral.
Another is forgetting to use π in the formula. However, π is crucial as it comes from the geometric fact that the area of a circle with radius r is πr^2.
Be careful about your limits of integration. They should coincide with the start and end points of the curve being rotated.