Volumes of Revolution

Exercises

Exercise 1

Derive the formula for the volume of a cone with base radius R and height H.

Figure 1(a)

cone with base radius R and height H

Hint

Solution

Exercise 2

Find the volume of a rugby football that has a center (circular cross section) radius of 95 mm and a length of 300 mm.

Hint

Solution

Exercise 3

Determine the volume of the given lightbulb (ignore the socket).

Figure 3(a)

lightbulb with coordinate axes drawn to show the x-axis representing the length and the y-axis representing the height of the bulb

Model the bulb using the piecewise function:

(Note that there is a small error where the functions join at x = 40 and x = 108, which will be ignored. The x units are in pixels, and 50 pixels = 1 inch.)

Figure 3(b)
top half of the lightbulb labeled with the length and width

This piecewise function was obtained by digitizing an image of the lightbulb in Mathematica and doing a curve fit to the digitized points. There are other software programs that can be used to do this same operation.

Solution

Exercise 4

The area between the functions and (shown in figure 4(a), below) is revolved around the y‑axis to form the water bowl shown in figure 4(b). What is the volume of material necessary to make the bowl, and how much water will it hold?

Figure 4(a)	Figure 4(b)

Solution