How to Accurately Calculate the Area Between Polar Curves Using Integrals

To find the area between two polar curves, \( r = f(\theta) \) and \( r = g(\theta) \), start by understanding the shape and range of each curve over a given interval. The area between them can be calculated by using the formula:

\( [
\text{Area} = \frac{1}{2} \int_{\alpha}^{\beta} \left( f(\theta)^2 – g(\theta)^2 \right) \, d\theta
] \)

Here’s how it works: \( f(\theta) \) represents the outer curve and \( g(\theta) \) the inner curve. When you square each function’s radius and subtract the inner from the outer, it captures the difference in area. Integrating this difference over the interval \([ \alpha, \beta ]\) accounts for the complete bounded area between curves.

This technique is useful in fields like physics and engineering, especially when calculating areas within circular fields or in motion analysis involving radial boundaries.

Consider two polar curves, \( r = 3 + \cos(\theta) \) and \( r = 1 \), over the interval \( \theta = 0 \) to \( \theta = \pi \). To find the area between these curves, use:

\([
\text{Area} = \frac{1}{2} \int_{0}^{\pi} \left( (3 + \cos(\theta))^2 – 1^2 \right) \, d\theta
]\)

Expanding and integrating each term gives the total area between the curves. This process efficiently captures the space enclosed by the curves over the specified interval.