Find the area bounded by the cardioid

[STEMucator]

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.

 

[LCKurtz]

It looks properly set up. Nice job. To answer your last question, I would draw the graph in polar coordinates to be sure. That is presumably no different than what you would do in any area problem in either rectangular or polar coordinates to check the limits and shape of the region. Polar graphs can surprise you by looping inside them selves or covering themselves more than once.