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22 Jan 2015

JC math tuition - How to find the volume generated by a curve?

One of the most problematic topics in A levels JC math is to find the volume generated when an area bounded by a curve is rotated. To do well in this topic, students not only have to be good at integration, they must also be able to imagine and visualize the volume of the object they are trying to find. Let’s take a look at a simple example to understand the concept behind this application of integration.

Suppose we want to rotate the shaded area about the x-axis through 2\pi (see diagram 1a), how can we do it?

Diagram 1a

Diagram 1a: Volume generated when shaded area is rotated about x-axis.

To understand how we can calculate the volume, we need to go back to our understanding of finding area bounded by a curve using the Riemann sum. To find the area bounded by the curve, we can divide the area into strips of rectangles. It is not difficult to understand that as the number of rectangles increases, the base of each rectangle, i.e. , \delta x will get smaller and tend to zero and the sum of the area of all the rectangles will be the approximated area bounded by the curve. This is called the Riemann sum, named after the German mathematician Bernhard Riemann.

We can now use the same concept to help us find the volume generated. If we rotate the rectangles about the x-axis, we will get cylinders as shown in diagram 1b.

Diagram 1b: Cylinders formed by rotating rectangles.

Diagram 1b: Cylinders formed by rotating rectangles.

The sum of the volume of the cylinders will provide us with an approximated volume of the solid generated. The approximation gets better as the number of cylinders increases and as \delta x \rightarrow 0. The volume of a cylinder is given by the formula \pi r^{2}h, where h is actually the base of the rectangle, \delta x, and r^{2} is actually the y^{2} value (see diagram 2).

 

Diagram 2: Volume of cylinders

Diagram 2: Volume of cylinder can be found using \pi r^{2} h.

Therefore, the approximated volume of the solid generated =\pi{}y_1^2\delta{}x+\pi{}y_2^2\delta{}x+\pi{}y_3^2\delta{}x+... As the number of rectangles \rightarrow{}\infty{} and \delta{}x\rightarrow{}0, the volume becomes \pi{}\int_{x_1}^{x_2}y^2dx.

Using the same logic, the formula for rotation about the y-axis is:

\pi{}\int_{y_1}^{y_2}x^2dy

However, the questions that students face in exams are usually not as straight forward. For example, find the volume of the shaded region, bounded by the curve 9{\left(x-1\right)}^2+y^2=9 and the line y=-3x+6 as shown in diagram 3, when it is rotated about the y-axis.

To successfully answer this question, students need to first visualize the object whose volume they are going to find. Diagram 4a shows the volume generated when the shaded region is rotated about the y-axis.

Diagram 4

Diagram 4

Students need to understand that this volume cannot be found in one step. They need to be able to visualize that they need to subtract the volume generated by the line (diagram 4c) from the volume generated by the curve (diagram 4b) in order to find the required volume.

Once they are able to visualize the objects, they should be able to form the necessary equation and solve the question. In my many years of teaching experience, I realized that once students are able to “see” and understand what they are dealing with, they usually will be able to solve the problem. 3-D computer animations will help students just starting to learn this topic fully grasp the concept from the beginning. However, students hardly have this kind of exposure in school. At numberskill Math Tuition, I use advanced mathematics software to help students learn math more effectively. I hope this article will help students struggling with this topic understand the topic better. For a detailed video solution of the example above, please view the video below.

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