Find the area inside r= 3 Cos[x] and outside of r= 1 + Cos[x]
Hello, camherokid!
I hope you made a sketch . . .Find the area inside and outside
is a circle with center and radius
is a cardioid with intercepts:
. . and "dimples in" to the origin from the left.
The polar formula for the area between two curves is: .
The curves intersect when: .
. . Hence, they intersect at: .
Therefore: .
We will use the formula:
where is the area between the curves and , and are the limits of integration (the points of intersection), is the outer curve, and is the inner curve.
First find the points of intersection:
this is where
we want to go from to , but we must go from a smaller angle to a bigger angle. changing to fixes this problem. so our area is given by:
EDIT: Beaten by Soroban! any way, that's ok. I'm a bit rusty on polar areas, so it's good to have a confirmation that I did the right thing. Soroban, can you check the other posts camherokid put up today and make sure I didn't make any mistakes