# Find the area bounded by the cardioid

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## Homework Statement

Find the area bounded by the cardioid $x^2 + y^2 = (x^2+y^2)^{1/2} - y$

## Homework Equations

Area of R = $\int \int_R dxdy = \int \int_{R'} |J| dudv$

J Is the Jacobian.

## The Attempt at a Solution

Switching to polars, x=rcosθ and y=rsinθ our region becomes $r^2 = r(1-sinθ) → r = 1-sinθ$
where 0 ≤ θ ≤ 2π.

Also, the Jacobian of polars is just r.

So our integral becomes :

$\int \int_R dxdy = \int \int_{R'} |J| dudv = \int_{0}^{2π} \int_{0}^{1-sinθ} r \space drdθ$

and using the identity $sin^2θ = (1/2)(1-cos(2θ))$, we can effectively evaluate it.

I have two concerns. The first concern is did I set this up right. My second concern which is more of a worry is how do I KNOW that 0 ≤ θ ≤2π without analytically showing it? It's leaving a sour taste that I'm not justifying it.

Last edited:

LCKurtz