# The prompt

Mathematical inquiry processes: Explore particular cases, conjecture, generalise, and prove. Conceptual field of inquiry: Functions, integration, limits of integration.

Andrew Blair designed the prompt for his Further Mathematics A-level class. It was inspired by a problem about the volume of revolution of an ellipse (see below). The students worked out the volume of revolutions about the x-axis and the y-axis.

Why are the volumes different? One member of the year 12 class explained that rotating the curve about the y-axis brought the 3-d shape further out of the page than the rotating it about the x-axis. The visualisation intrigued the class.

The next lesson started with the prompt. After coming to a common understanding of the statement, students started to speculate about whether it could be true. Considering the first quadrant only (x > 0, y > 0), the linear function f(x) = x would give equal areas for any limits. What is the relationship between the volumes for the functions f(x) = nx (where n is a positive integer)?

Other students suggested looking at quadratic and cubic functions, but doubted if they could possibly satisfy the conditions in the prompt.

June 2024

# Lines of inquiry

## 1. Exploring linear functions

Adam and Sophie posed a question about the volumes of revolution using a straight line: What is the ratio between the volumes as you increase the gradient of a straight line? They used two approaches - integrating the linear function and calculating the volume of a cone.

The ratio of the volume of revolution around the x-axis to the volume of revolution around the y-axis turned out to be m:1, where m is the gradient of the line.

In their feedback to the class, the students explained the result by considering the formula for the volume of a cone. The radius and height are switched when considering the two revolutions and, as the radius is squared, it has a greater impact on the volume.

2. Conjecture and spatial reasoning

In their initial response to the prompt, Muhammad and Jack considered the graph of the equation y = x2 + 2. They came up with a conjecture: "An increase in c would not change the volume of revolution around the y-axis but would around the x-axis."

The rest of the class agreed with the conjecture if the limits on the x-axis (-2 and 2) remained the same. The two students went on to test their conjecture (see below).

### Follow-up question

During the inquiry, Muhammad posed a follow-up question: Is there a value of c for which the volumes of revolution are equal?

As the volume of revolution about the y-axis is always the same, the question reduced to finding the value of c that gives a volume of revolution about the x-axis of 8 pi. There are two solutions and, in both cases, the total volume is made up of three separate solids.

### Exploring solids

Another line of inquiry develops from comparing the volume of revolution of the curve of y = x2 + 2 about the x-axis and the solid formed by rotating the curves of y = x2 + 2 and y = x2 - 2 about the y-axis. Using the limits of -2 and 2 on the x-axis (see the illustration), the volume around the x-axis is greater. The class felt that that would always be the case unless the curve could be 'flattened' by making the coefficient of x2  less then one.

3. A cubic function

Taisei and Batu set out to explore the limits for the volumes of revolution for the function f(x) = x3. They aimed to find limits on the x-axis, 0 and x, such that the limits on the y-axis, 0 and f(x), gave the same volume of revolution.

Taiyo found x for the case when x = f(x); Bartu (picture below) solves the problem completely by using x and x3 respectively for the limits. The volumes of revolution are equal when x is (√105)/5.