Chapter 9: Q. 46 (page 756)

Use Theorem 9.13 to show that the area of the circle defined by the polar equation $r = 2 a \cos θ$ is $π a^{2}$ .

Short Answer

Expert verified

The area of the circle is $π a^{2}$ .

Step by step solution

Step 1. Given Information

The circle is bounded by the curve $r = 2 a \cos θ$ .

Step 2. Use the formula for area of a region bounded by a curve

The region is bounded by the curve $r = 2 a \cos θ$ .
The region is a circle, so the boundary arcs will be $θ = 0 to θ = 2 π$ .
But the curve is traced twice in this region.
So, the area of the region will be calculated as follows:

$\begin{array}{rcl} \frac{1}{2} \times \frac{1}{2} \int_{0}^{2 π} 4 a^{2} \cos^{2} θ d θ & = & a^{2} \int_{0}^{2 π} \cos^{2} θ d θ \\ = & a^{2} \int_{0}^{2 π} \frac{1 + \cos (2 θ)}{2} d θ \\ = & a^{2} \times \frac{1}{2} (\int_{0}^{2 π} 1 d θ + \int_{0}^{2 π} \cos (2 θ) d θ) \\ = & \frac{1}{2} a^{2} (2 π + 0) \\ = & π a^{2} \end{array}$

Hence, the area of the curve is $π a^{2}$ .

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Use Theorem 9.13 to show that the area of the circle defined by the polar equation $r = 2 a \cos θ$ is $π a^{2}$ .

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Use the formula for area of a region bounded by a curve

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Use Theorem 9.13 to show that the area of the circle defined by the polar equationr=2acosθisπa2.

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Use the formula for area of a region bounded by a curve

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

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Study anywhere. Anytime. Across all devices.

Use Theorem 9.13 to show that the area of the circle defined by the polar equation $r = 2 a \cos θ$ is $π a^{2}$ .