Question

Find the area of the region bounded by the graph of the polar equation using (a) a geometric formula and (b) integration. $$ r=3 \cos \theta $$

Short answer

The area of the region bounded by the polar equation r=3cos(\(\theta\)) is \(2.25\pi\) using geometric formula and \(4.5\pi\) using integration.

Step-by-step solution

Solving Using Geometrics

First, the formula for the area of a circle, \(A = \pi r^2\), should be used. The circle's radius is half of the maximum value of r, given by r=3cos(\(\theta\)), so the radius is r = 1.5. Therefore, the area A = \(\pi * (1.5)^2\).

Solving Using Integration

Alternatively, another approach is using integration. The area of a polar graph is given by \(0.5* \int_{0}^{\pi} (3cos(\(\theta\)))^2 d \theta\). Integrating this equation gives the area under the curve defined by the polar equation from 0 to pi.

Calculating the Integral

Evaluating the integral gives an alternate solution for the area. The integral works out to \(0.5* \int_{0}^{\pi} 9cos^2(\(\theta\))d \theta\), which can be solved to \(4.5\pi\).