Finding the Area Inside Polar Equations

[apphysicsgirl]

1. Find the area of the region described:
a) inside one loop of the lemniscate r^2=4cos(2theta)
b) inside the six-petaled rose r^2=2sin(3theta)
2. A=integral [1/2 r^2 dtheta]
Are there any easy ways to determine the integration bounds? (without graphing)
Our textbook doesn't give any examples like this.

 

[Mark44]

I don't think there's an easy way that doesn't involve graphing something. For the first one, sketch a graph of y = 4cos(2x) and for the second one, sketch a graph of y = 2sin(3x).

These are not polar graphs, but they can give you some insight into what the corresponding polar graphs look like.

For y = 4cos(2x), the period is pi, so it's pretty easy to find an interval on the x-axis that corresponds to one loop of the polar curve. Note that you don't want any y values that are negative, since your polar curve involves r^2.