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 graph of the polar equation is \( 4.5 \pi \) units.

Step-by-step solution

Understanding the Polar Equation

The polar equation given \( r = 3 \cos \theta \) represents a circle with a radius of 3 units. The value of \( r \) depends on the value of \( \theta \). When \( \theta = 0 \), \( r = 3 \), and as \( \theta \) approaches \( \pi/2 \), \( r \) goes to 0.

Calculate the Area Geometrically

The area of a circle is given by the formula \( A = \pi r^2 \). Since this graph only depicts the right half of the circle, the area will be half of a full circle's area. Hence, the formula becomes \( A = \pi/2 (3)^2 = 4.5 \pi \).

Calculate the Area Using Integration

The formula to calculate the area inside a polar curve from \( \theta=0 \) to \( \theta=\pi/2 \) is \( A = \int^{ \pi/2}_{0} \frac{1}{2} (3\cos\theta)^2 d\theta \). After performing the integration and substitution, you will also get \( A = 4.5 \pi \).