Mastering Mathematics Smartly
by Wee Wen Shih

A unique self-help website that provides comprehensive coverage of mathematics at A-level & beyond, written in a student-friendly style.

Solids of revolution

This document compares between solids of revolution about x- and y-axes.

This website features interactive examples with detailed steps in calculating volumes. Please ignore examples 4 and 5 which are not in H2 Mathematics syllabus.

Knowledge of integration techniques, e.g. substitution, by-parts, etc. are necessary when evaluating integrals.

Additional practical tips:
1. Revolve a quadrant of circle: a hemisphere is produced.

2. Revolve a semi-circle: a sphere is produced.

3. Revolve a triangular region: a cone is produced.

4. Revolve a square or rectangular region: a cylinder is produced.

A worked example

The curve C has equation given by x^2 + 3xy = 4. The region bounded by C, the line y = 1 and the axes is rotated through 4 right angles about the y-axis. Find the volume of revolution.

Approach:
1. Identify the region by a quick sketch, with the aid of GC. Note that the equation may be rewritten as y = \frac{4 - x^2}{3x}.



2. The volume is then given by \pi\int_{0}^{1} x^2 \text{d}y.

We will need to express x in terms of y (an important skill to be kept in one's mathematical arsenal!), i.e.
x^2 + (3y)x - 4 = 0 \\ \Rightarrow x = \frac{-3y \pm \sqrt{(3y)^2 - 4(1)(-4)}}{2(1)} \\ \Rightarrow x = \frac{-3y \pm \sqrt{9y^2 + 16}}{2}.

From the sketch, we see that x > 0, so we can conclude that x = \frac{-3y + \sqrt{9y^2 + 16}}{2} is our desired expression.

We are now ready to evaluate our integral:
Volume =\frac{\pi}{4}\int_{0}^{1} \left(-3y + \sqrt{9y^2 + 16}\right)^2 \text{d}y
=\frac{\pi}{4}\int_{0}^{1} \left[18y^2 -6y \sqrt{9y^2 + 16} + 16\right] \text{d}y, after expansion and simplification
=\frac{19\pi}{9}\text{units}^3.

What if we revolve the region R about the x-axis and we want to find its volume? It is easy to see that the volume is made up of a cylinder (with radius 1 unit and height 1 unit) and the integral \pi \int_{1}^{2} \left(\frac{4}{3x} - \frac{x}{3}\right)^2 \text{d}x, which is easy to evaluate after expanding and simplifying it.