Rectification - Polar Curves - Problem 1 - Rectification - Engineering Mathematics - 2

TL;DR

This video explains how to find the length of a cardioid using the concept of rectification, providing step-by-step instructions and equations.

Transcript

hello students so now we are gonna see a numerical which is based on the polar cost so uh i'll take one polar curve and we'll find out length of that curve using the concept of rectification so now here we have to find out the length of cardioid which is r equal to a into 1 minus cos theta like outside the circle are equal to a cos theta so guys be... Read More

Key Insights

• ❓ The length of a cardioid curve can be found using the concept of rectification.
• ⭕ Drawing the cardioid curve and a circle with their respective equations helps visualize the problem.
• 😥 The limits of integration are determined by finding the points of intersection between the circle and the cardioid curve.
• ✖️ The length of the curve below the x-axis is symmetric to the length above, so it can be multiplied by 2 to find the total length.

Questions & Answers

Q: What is the equation for the cardioid curve?

The equation for the cardioid curve is r = a(1 - cos(theta)).

Q: How do you find the length of the cardioid curve outside the circle?

To find the length of the cardioid curve outside the circle, you need to use the formula for rectification and integrate the equation for the curve.

Q: How do you determine the limits of integration for finding the length of the curve?

The limits of integration can be determined by finding the points of intersection between the cardioid curve and the circle. These points will define the start and end of the curve.

Q: Why do you multiply the calculated length by 2?

The length of the curve outside the circle is symmetric to the length below the x-axis. Multiplying the calculated length by 2 accounts for this symmetry and gives the total length of the cardioid curve.

Summary & Key Takeaways

• The video demonstrates how to find the length of a cardioid curve using the equation r = a(1 - cos(theta)).

• It explains the process of drawing the cardioid curve and a circle with equations r = a(1 - cos(theta)) and r = a cos(theta) respectively.

• The video then introduces the concept of rectification and demonstrates how to apply the formula to find the length of the cardioid curve.