Area of Surfaces of Revolution with Polar Equations - Calculus 2

The final application of calculus within the polar coordinate system is calculating the area of surfaces of revolution formed by polar curves via integration. This includes surfaces of revolution formed by revolving polar curves around the polar axis (equivalent to the x-axis) and the vertical axis of θ=π/2 (equivalent to the y-axis).

Previously we already saw how to calculate area of surfaces of revolution in the rectangular coordinate system when they are created by curves in rectangular form and parametric form. In fact, by revisiting the formulas/definite integrals for calculating the area of surfaces of revolution in the rectangular coordinate system, we can see how to find similar formulas to calculate the area of surfaces created by polar curves in the polar coordinate system.

The main idea to keep in mind is that calculating the area of these surfaces of revolution involves taking the circumference of a representative circular cross-section of the surface and multiplying it by the arc length of the curve that is being revolved around an axis. We already know how to find the arc length of polar curves (working in terms of θ), so if we can represent the circumference of a representative circular cross-section in terms of θ, we can easily create definite integrals to represent the area of the surfaces of revolution.

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:

• The definite integrals that calculate area of surfaces of revolution formed by polar curves
• How to find the area of surfaces of revolution formed by revolving around the polar axis
• How to find the area of surfaces of revolution formed by revolving about θ= π/2 

I hope you find this video to be helpful!

 -Josh

P.S. This is the final lesson in my Calculus 2 series! There will be an examples video to accompany this lesson, but after that my coverage of Calc 2 is complete. If you have been following along or have been using these videos to help you learn Calc 2, I want to give you a big thank you for watching the videos and reading these posts! I hope you found all the videos up to this point to be helpful! I am excited to share what is next at JK Mathematics, but for now enjoy this last topic in Calc 2!