Exercise 26

A cone with base radius $a$ and height $h$ can be realized as the surface defined by $z = \frac{h}{a} \sqrt{x^2+y^2}$, where $a$ and $h$ are positive.

1. Find a parameterization of the cone described by $z = \frac{h}{a} \sqrt{x^2+y^2}$. (Hint: Compare to the parameterization of a cylinder as seen in Activity.)

2. Set up an iterated integral to determine the surface area of this cone.

3. Evaluate the iterated integral to find a formula for the lateral surface area of a cone of height $h$ and base $a$.