Area Calculation for Circle and Cardioid Using Double Integrals

[stolencookie]

Homework Statement

r=1 and r=1+cos(theta), use a double integral to find the area inside the circle r=1 and outside the cardioid r=1+cos(theta)

Homework Equations

The Attempt at a Solution

I am confused on the wording and how to set it up. I tried setting it up by setting theta 0 to pi. and r as 1 to 1+cos(theta). I used r drd(theta) as the equation to use.

 

[BvU]

Homework Statement

r=1 and r=1+cos(theta), use a double integral to find the area inside the circle r=1 and outside the cardioid r=1+cos(theta)
...

I am confused on the wording and how to set it up.

Make a picture !

 

[stolencookie]

Make a picture !
View attachment 215925

I did make a picture I am confused by the little piece of the cardoid that isn't in the first quadrant.

 

[Ray Vickson]

I did make a picture I am confused by the little piece of the cardoid that isn't in the first quadrant.

In the whole plane, what is the region outside the cardioid? What is the region inside the circle? What is the intersection of those two regions?

 

[Mark44]

I tried setting it up by setting theta 0 to pi

As shown in BvU's graph, the region of integration is entirely on the left side of the vertical axis. What is $\theta$ at the upper intersection point? At the lower intersection point? There is also some symmetry you can take advantage of.

 

[BvU]

little piece of the cardoid that isn't in the first quadrant

Ambiguous -- in the picture a small piece is missing because I simply didn't grab the full $\theta$ range for the red curve