Volume of revolution inquiry
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.
