Area of Surfaces of Revolution - Calculus 2

In previous lessons we learned how to find the volume of solids of revolution using definite integrals via the disk, washer, and shell methods. Similarly, we can determine the area of surfaces of revolution using definite integrals.

But what is a surface of revolution? Well they are similar to solids of revolution, with one major difference. The difference is that a solid of revolution is formed by revolving a region beneath a curve around an axis of revolution, but a surface of revolution is formed by revolving just the curve around the axis of revolution. As such, these surfaces have no volume, and could be viewed as the outside of its solid of revolution counterpart.

In order to calculate the area of these surfaces, we can use definite integrals in a similar manner to how we used them to calculate volume. However in order to do so, we will need to revisit the arc length formula from the previous lesson of Calculus 2. This is because the surfaces of revolution will have circular cross sections, and so the total area can be found by multiplying the circumference of those circles by the length of the curve that forms the surface.

In this lesson, you will learn:

• What surfaces of revolution are and how to find their area using arc length and a definite integral
• How to calculate the area of surfaces of revolution formed by revolving a curve around the x-axis
• How to calculate the area of surfaces of revolution formed by revolving a curve around the y-axis

I hope you find this video to be helpful!

-Josh