How to Find the Area Under a Parametric Curve

TL;DR

To find the area between a parametric curve and the x-axis, set up the integral using vertical rectangles where the area is calculated by multiplying the vertical distance (Y) by the small change in x (DX). Convert parametric equations to Cartesian form if necessary, and integrate from the starting to the ending T-values.

Transcript

okay this video we're gonna see how to find the areas with parametric equations and we are still talking about the same paper entry questions from the previous videos and this time we just want to focus on this portion of the curve from TS 1 up to 4 and we want the area of this rigid notice I have the Cartesian equation down here first used as well... Read More

Key Insights

• ☺️ The area between a parametric curve and the x-axis can be found using vertical rectangles.
• 😫 The integral for the area is set up by substituting the parametric equations and integrating.
• 💁 Parametric equations in Cartesian form need to be converted to parametric form before finding the area.
• 😰 DX is replaced by X prime of T DT when the equation is in terms of T.

Questions & Answers

Q: How can the area between a parametric curve and the x-axis be found?

The area can be found by multiplying the y-value of the curve by the small change in x (DX). This integral is set up by substituting the parametric equations into the formula and integrating.

Q: Can the process be applied to parametric equations in Cartesian form?

No, the process specifically deals with parametric equations. If you have Cartesian equations, you need to convert them into parametric form before finding the area.

Q: Why is DX replaced by X prime of T DT in the second part of the video?

In the second part, the equation is in terms of T rather than X. So, DX is replaced by DX DT, which is obtained by differentiating the X equation with respect to T.

Q: How do you find the area using parametric equations in the second part?

The area is found by integrating the product of Y of T and X prime of T with respect to T, from the starting T-value to the ending T-value.

Summary & Key Takeaways

• The video teaches how to find the area between a parametric curve and the x-axis using vertical rectangles.

• It explains that the area can be calculated by multiplying the vertical distance (Y) by the small change in the x-distance (DX).

• The process involves setting up the integral with the starting and ending x-values (or T-values) and substituting the parametric equations into the formula.