Area Under Parametric Curves - Calculus 2

In Calculus 1, we are introduced to the concept that a definite integral calculates the area under a curve between two values of x. Up to this point, the area that we wanted to calculate was beneath curves that were always defined with functions in terms of x and y, or rectangular equations. But what if the curve that we want to calculate the area under is a plane curve/parametric curve represented with parametric equations? How can we calculate the area?

Well, the process is actually not too complicated:

To calculate area under a parametric curve, we begin by sketching a graph of the curve, and then use our Calculus 1 skills to set up an integral in terms of x to represent the area that we want to calculate. After that integral is set up, we want to convert it to be in terms of t, or whatever parameter the parametric equations are defined with. Once the integral is converted, we can simplify and solve the integral to calculate the area.

Now, in some situations it may be more reasonable to work in terms of x instead of the parameter to calculate the area. For example, if the parametric equations are fairly simple and easily converted into the corresponding rectangular equation that can be solved for y (in terms of x), and that equation is easily integrated, then it may be advantageous to work in terms of x rather than the parameter.

However, the majority of curves represented with parametric equations will not meet all those requirements. Many parametric equations have complicated corresponding rectangular equations that make it difficult or nearly impossible to solve for y. And even when solving for y is possible, the resulting equation in terms of x is probably not easily integrated.

As a result of this, you are going to find that in most cases, calculating area under parametric curves will be easier when working in terms of the parameter. This is especially true for parametric curves represented with parametric equations involving trigonometric functions.

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 set up an integral in terms of the parameter to represent area under a parametric curve
• How to solve integrals in terms of the parameter to calculate area
• When/Why working in terms of the parameter is advantageous

I hope you find this video to be helpful!

-Josh