Polar coordinates

Cartesian vs Polar

\begin{matrix} (r, θ) \to (x, y) & (x, y) \to (r, θ) \\ \begin{matrix} x & = & r cos θ \\ y & = & r sin θ \end{matrix} & \begin{matrix} r^{2} & = & x^{2} + y^{2} \\ tan θ & = & \frac{x}{y} \end{matrix} \end{matrix}

Let $r = r (θ)$ be continuous and non-negative on $θ \in [α, β]$ Then the enclosed area $𝒜$ is

𝒜 = \frac{1}{2} \int_{α}^{β} {[r (θ)]}^{2} ⅆ θ

Let $r = r (θ)$ be differentiable on $θ \in [α, β]$ , the arc length $ℒ$ of the corresponding parametric curve is

ℒ = \int_{α}^{β} \sqrt{{[r (θ)]}^{2} + {[r^{'} (θ)]}^{2}} ⅆ θ

r (θ) = 2 + 2 sin θ

Area