Area by Double Integration Polar Coordinates Problem 5

TL;DR

Calculate the area of a region bounded by a limacon and a circle using double integration.

Transcript

hello everyone so in this session we'll see another problem on area by double integration for polar coordinates so the question is as such which is find the area of the region that lies inside r equal to 3 plus 2 sine theta and outside r equal to 2. so we have got two curves here first curve is r equal to 3 plus 2 sine theta which is a limicon and ... Read More

Key Insights

• 😥 The area calculation involves finding the intersection points, determining the range of theta, and establishing the limits for r.
• 🪜 Depending on the desired region, theta can be varied by adding 2pi or starting from a negative angle.
• 🫡 The area formula involves double integration with respect to theta and r, using the corresponding limits.
• 🍉 The integration process involves simplification, term expansion, and applying trigonometric identities.
• 🤨 The final result for the area is given in terms of pi and the square root of 3.

Questions & Answers

Q: What curves are involved in finding the area?

The curves involved are a limacon given by r = 3 + 2sin(theta) and a circle given by r = 2.

Q: How do you find the intersection points between the two curves?

To find the intersection points, set 3 + 2sin(theta) equal to 2 and solve for sin(theta). The solutions are theta = pi/6 and 11pi/6.

Q: What is the range of theta for the desired region?

The desired region lies between theta = -pi/6 and theta = 7pi/6, taking into account a full rotation of 2pi.

Q: How do you determine the limits for r?

The inner limit is given by the circle r = 2, and the outer limit is given by the limacon r = 3 + 2sin(theta).

Summary & Key Takeaways

• The content discusses the problem of finding the area of a region bounded by a limacon and a circle in polar coordinates.

• It explains the process of determining the intersection points between the curves and finding the range of theta.

• The video then shows how to establish the limits for the radial variation and perform double integration to calculate the area.