Area by Double Integration Polar Coordinates Problem 2

TL;DR

The video explains how to find the area enclosed by a circle and a cardioid in polar coordinates using double integration.

Transcript

hello in this session we are going to discuss another problem on area by double integration for polar coordinates so let us see the question first so the question says find by double integration the area lying inside the circle and outside the cardioid which is given like both the equations are given so as per that we need to find the area enclosed... Read More

Key Insights

• 🐻‍❄️ The problem involves finding the area enclosed by a circle and a cardioid in polar coordinates.
• ⭕ The equation for the circle is r = asin(theta), and the equation for the cardioid is r = a(1 - cos(theta)).
• ⭕ The common area between the circle and the cardioid is the region inside the circle but outside the cardioid.
• 🫡 The integration involves integrating the equation for the common area with respect to theta and applying the appropriate limits.
• ⏫ The double integration results in the formula for the area: a^2/2*(2 - pi/2) or 2a^2 - (pi*a^2)/4.
• ❓ The rearranged formula is also valid: a^2/2*(1 - pi/4), depending on the given options.

Questions & Answers

Q: What is the problem being discussed in the video?

The problem involves finding the area inside a circle and outside a cardioid using double integration in polar coordinates.

Q: How are the equations for the circle and the cardioid represented in polar coordinates?

The equation for the circle is r = asin(theta), and the equation for the cardioid is r = a(1 - cos(theta)).

Q: How is the common area between the circle and the cardioid determined?

The common area is the region that is inside the circle but outside the cardioid, as shown in the video.

Q: What are the limits for the angle (theta) and the radius (r) in the double integration?

The angle (theta) varies from 0 to pi/2, and the radius (r) varies from a*(1 - cos(theta)) to a*sin(theta).

Summary & Key Takeaways

• The video discusses a problem on finding the area inside a circle and outside a cardioid using double integration in polar coordinates.

• It explains the equations for the circle and the cardioid and identifies the common area between them.

• The video demonstrates the process of setting up the double integration and integrates the equations to find the enclosed area.