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Chapter 9: Problem 4

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

Expert verified
The area of the region bounded by the graph of the polar equation is \( 4.5 \pi \) units.

Step by step solution

01

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.
02

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 \( A = \pi/2 (3)^2 = 4.5 \pi \).
03

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 \( A = 4.5 \pi \).

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