Chapter 6.4, Problem 50E

Comparing American and rugby union footballs An ellipse centered at the origin is described by the equation $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$, where a and b are positive constants. If the upper half of such an ellipse is revolved about the x-axis, the resulting surface is an ellipsoid.

1. a. Use the washer method to find the volume of an ellipsoid (in terms of a and b). Check your work using the shell method.
2. b. Both American and rugby union footballs have the shape of ellipsoids. The maximum regulation size of a rugby union football corresponds to parameters of a = 6 in and b = 3.82 in, and the maximum regulation size of an American football corresponds to parameters of a = 5.62 in and b = 3.38 in. Find the volume of each football.
3. c. Fill in the blank: At their maximum regulation sizes, the volume of a rugby union football has approximately _____ times the volume of an American football.