Chapter 9: Q. 32 (page 775)

Areas of regions bounded by polar functions: Find the areas of the following regions.
The area bounded by the function $r = a \cos θ$ , where $a$ is a positive constant.

Short Answer

Expert verified

The area bounded by the function is $\frac{{πa}^{2}}{2} {unit}^{2}$

Step by step solution

Given information.

The given function is $r = a \cos θ$ ,where a is a positive constant.

Area

The area of the curve is given by,

$A = \frac{1}{2} \int_{α}^{β} r^{2} d θ A = \frac{1}{2} \int_{0}^{2 π} {(a \cos θ)}^{2} d θ A = \frac{1}{2} \int_{0}^{2 π} (a^{2} \cos^{2} θ) d θ A = \frac{a^{2}}{2} \int_{0}^{2 π} (\frac{1 + \cos 2 θ}{2}) d θ A = \frac{a^{2}}{4} \int_{0}^{2 π} (\frac{1 + \cos 2 θ}{2}) d θ A = \frac{a^{2}}{4} {[θ + \frac{\sin 2 θ}{2}]}_{0}^{2 π} A A = \frac{a^{2}}{4} {[2 π + \frac{\sin 4 π}{2}]}_{0}^{2 π} A = \frac{{πa}^{2}}{2} {unit}^{2}$

The area of the function is $A = \frac{{πa}^{2}}{2} {unit}^{2}$

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Areas of regions bounded by polar functions: Find the areas of the following regions.
The area bounded by the function $r = a \cos θ$ , where $a$ is a positive constant.

Short Answer

Step by step solution

Given information.

Area

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Areas of regions bounded by polar functions: Find the areas of the following regions.The area bounded by the function r=acosθ, where a is a positive constant.

Short Answer

Step by step solution

Given information.

Area

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Applied Mathematics

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Probability and Statistics

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Study anywhere. Anytime. Across all devices.

Areas of regions bounded by polar functions: Find the areas of the following regions.
The area bounded by the function $r = a \cos θ$ , where $a$ is a positive constant.