Section 7.4 Area and Arc Length in Polar Coordinates
References.
- Calculus, Early Transcendentals by Stewart, Section 10.4.
Areas of Regions Bounded by Polar Curves.
A polar curve \(r = f(\theta)\) typically encloses a region inside the curve, \(r < f(\theta)\) rather than below it. That is, we are often interested in the region between the curve and the pole (origin), rather than between the curve and a horizontal axis.
Thus, the strategy for finding area as the integral of infinitesimal fragment or area differential \(dA\) will be based on looking at the thin region that lies between the curve and the pole over a narrow range of polar angle values \(d\theta\text{:}\) a very thin sector of radius \(r = f(\theta)\) and angular extent \(d\theta\)
A circular sector of radius \(r\) covering angle \(\theta\) has area \(\frac{1}{2}r^2 \theta\text{;}\) thus this infinitesimal sector has area described by the area differential
\begin{equation}
dA = \frac{1}{2}r^2 d\theta = \frac{1}{2}[f(\theta)]^2 d\theta \tag{7.4.1}
\end{equation}
If you prefer using approximations with small finite pieces, the range of angles \(\theta_i \leq \theta \leq \theta_i + \Delta \theta\) gives a region that is approximately a sector, with area approximation
\begin{equation*}
\Delta A_i \approx \frac{1}{2}[f(\theta_i^*)]^2 \Delta \theta
\quad \text{ for some angle }\theta_i^* \text{ in }[\theta_i, \theta_i+\Delta\theta]
\end{equation*}
The transition from finite increments to infinitesimal ones turns this into the exact formulas seen above,
\begin{equation*}
dA = \frac{1}{2}r^2 d\theta = \frac{1}{2}[f(\theta)]^2 d\theta.
\end{equation*}
The familiar argument with limits and the FTC then shows that the area of the “sector” inside the curve \(r = f(\theta)\text{,}\) \(a \leq \theta \leq b\) is
\begin{equation}
A = \int dA = \frac{1}{2} \int_{\theta=a}^br^2 d\theta = \frac{1}{2} \int_{\theta=a}^b [f(\theta)]^2 d\theta.\tag{7.4.2}
\end{equation}
Arc Length in Polar Curves.
The arc length of a polar curve \(r = f(\theta)\text{,}\) \(a \leq \theta \leq b\) is given by simply using the results for a parametric curve applied to
\begin{equation*}
x = r \cos\theta, \quad y = r \sin\theta.
\end{equation*}
The arc length differential for this polar curve,
\begin{equation*}
ds = \sqrt{\left(\frac{dx}{d\theta}\right)^2 + \left(\frac{dy}{d\theta}\right)^2} d\theta,
\end{equation*}
simplifies nicely when combined with the formulas for the derivatives
\begin{equation*}
\begin{split}
\frac{dx}{d\theta} & = \frac{dr}{d\theta} \cos\theta + r \sin\theta
\\
\frac{dr}{d\theta} & = \frac{dr}{d\theta} \sin\theta - r \cos\theta
\end{split}
\end{equation*}
to give
\begin{equation}
ds = \sqrt{r^2 + \left(\frac{dr}{d\theta}\right)^2} d\theta,\tag{7.4.3}
\end{equation}
and thus
\begin{equation}
L = \int ds = \int_{\theta=a}^b \sqrt{r^2 + \left(\frac{dr}{d\theta}\right)^2} d\theta.\tag{7.4.4}
\end{equation}
Study Guide.
Study Calculus Volume 2, Section 7.4 2 ; in particular
- Theorems 6 and 7
- Examples 16 and 18
- Checkpoints 15 and 17
- and one or several exercises from each of the following groups: 188–194, 201–206, 214–217, 218–222.
openstax.org/books/calculus-volume-2/pages/7-4-area-and-arc-length-in-polar-coordinatesopenstax.org/books/calculus-volume-2/pages/7-4-area-and-arc-length-in-polar-coordinates
