Areas in Polar Coordinates¶
Suppose $r = \rho(\theta)$ is a polar function. We wish to compute the area enclosed between the polar graph of $r = \rho(\theta)$ and the pole, as depicted by the shaded region in the graph below.
Video Lecture
Polar Area Formula
$\text{Area}\ = \displaystyle \int_\alpha^\beta \frac{1}{2} \big[ \rho(\theta) \big]^2\, d\theta$.
1. Example¶
Compute the area of one leaf of the four-leaved rose, $r = \cos(2\theta)$.
Check your answer
$$\frac{\pi}{8}$$
Partial video solution
2. Example¶
Compute the area of the region that is bounded outside of the cardioid $r = 1 - \sin\theta$ but inside of the circle $r = \cos\theta$.
Check your answer.
$$\frac{4-\pi}{4}$$
Partial video solution
Parametrizing Polar Curves¶
Suppose $r = \rho(\theta)$ is a polar function. We wish to represent the polar graph of this function as a parametric curve in the Cartesian plane.
Recall the polar transformation,
$$
T(r,\theta) = \begin{cases} x = r\cos\theta, \\ y = r\sin\theta. \end{cases}
$$
Since $r = \rho(\theta)$, the curve may be parametrized by
$$
f(\theta) = \begin{cases} x(\theta) = \rho(\theta)\cos(\theta), \\
y(\theta) = \rho(\theta)\sin(\theta).
\end{cases} \tag{$*$}
$$
3. Example¶
Find the polar parametrization of the cardioid, $r = \cos\theta + 1$.
Check your answer.
$$f(\theta) = \begin{cases}
x(\theta) = \cos^2\theta + \cos \theta \\
y(\theta) = \sin\theta\cos\theta + \sin\theta.
\end{cases}$$
Arc Length in Polar Coordinates¶
Recall that the arc length element for a parametric curve is given by
$$
ds = \sqrt{\dot{x}^2 + \dot{y}^2}\, d\theta,
$$
where $\theta$ is the parameter.
4. Exercise¶
Suppose $r = \rho(\theta)$ is a polar function. Use the parametrization $(*)$ to write out the arc length element $ds$ in terms of $\rho$ and $\dot{\rho}$.
Check your answer.
$$\displaystyle ds = \sqrt{\rho^2 + \dot{\rho}^2\,}\, d\theta$$
Video solution
5. Example¶
Compute the arclength of the cardioid, $r = \cos\theta + 1$.
Check your answer.
It's usually a good idea to start by sketching the curve.
The arclength of this curve is given by
$$s = \int_0^{2\pi} \sqrt{2 + 2\cos\theta}\,d\theta = 8.$$
Partial video solution
Discussion¶
Questions? You can ask them below.