- #1
- 2,076
- 140
Homework Statement
Find the area bounded by the cardioid [itex]x^2 + y^2 = (x^2+y^2)^{1/2} - y[/itex]
Homework Equations
Area of R = [itex]\int \int_R dxdy = \int \int_{R'} |J| dudv[/itex]
J Is the Jacobian.
The Attempt at a Solution
Switching to polars, x=rcosθ and y=rsinθ our region becomes [itex]r^2 = r(1-sinθ) → r = 1-sinθ[/itex]
where 0 ≤ θ ≤ 2π.
Also, the Jacobian of polars is just r.
So our integral becomes :
[itex]\int \int_R dxdy = \int \int_{R'} |J| dudv = \int_{0}^{2π} \int_{0}^{1-sinθ} r \space drdθ[/itex]
and using the identity [itex]sin^2θ = (1/2)(1-cos(2θ))[/itex], 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: