Area of Surfaces of Revolution with Parametric Equations - Calculus 2

Previously in Calculus 2 (Lesson 7), we looked at how to use definite integrals to calculate the area of surfaces of revolution, which are formed by revolving a curve around either the x-axis or y-axis. Whenever we calculated the area of these surfaces of revolution, we always worked in terms of x because the curves that formed the surfaces had always been defined with functions that were in terms of x and y (rectangular equations). However, now we are going to look at how to calculate the area of the same surfaces of revolution, but where the curves that form the surfaces are defined with parametric equations instead of rectangular equations.

To calculate the area of surfaces of revolution formed by curves represented by parametric equations, we will need to work terms of the parameter from the parametric equations (typically t). Therefore we need to develop formulas in terms of the parameter that represent the area of the surfaces formed by revolving the curve around the x-axis and the y-axis. We can do this by reworking the original formulas for area of surfaces of revolution from being in terms of x to being in terms of the parameter. Once we determine these formulas, we can apply them to any scenario where we want to calculate the area of a surface of revolution formed by a parametric curve (provided that the curve is smooth, and is without any gaps, sharp points, or points where the curve intersects itself). 

We discuss all of this and more in this new lesson of Calculus 2. Let's get to it!

In this lesson, you will learn:

• How to calculate the area of surfaces of revolution formed by parametric curves revolved around the x-axis
• How to calculate the area of surfaces of revolution formed by parametric curves revolved around the y-axis

I hope you find this video to be helpful!

-Josh