Find the area inside r= 3 Cos[x] and outside of r= 1 + Cos[x]

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- Jul 21st 2007, 04:30 PMcamherokidPolar Coordinate 03
Find the area inside r= 3 Cos[x] and outside of r= 1 + Cos[x]

- Jul 21st 2007, 05:43 PMSoroban
Hello, camherokid!

Quote:

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: .

- Jul 21st 2007, 05:52 PMJhevon
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 - Jul 21st 2007, 06:00 PMcurvature