Chapter 9: Q. 33 (page 775)

Areas of regions bounded by polar functions: Find the areas of the following regions. The area bounded by one petal of $r = \cos 2 θ$

Short Answer

Expert verified

The area bounded by one petal is $\frac{π}{8} {unit}^{2}$

Step by step solution

Given information

The curve is $r = \cos 2 θ$

Area

The area bounded by the curve is,

$A = \frac{1}{2} \int_{α}^{β} r^{2} d θ A = \frac{1}{2} \int_{0}^{\frac{π}{4}} {(\cos 2 θ)}^{2} d θ A = \frac{1}{2} \int_{0}^{\frac{π}{4}} (\cos^{2} 2 θ) d θ A = \frac{1}{2} \int_{0}^{\frac{π}{4}} \frac{(1 + \cos 4 θ)}{2} d θ A = \frac{1}{2} {[θ + \frac{\sin 4 θ}{4}]}_{0}^{\frac{π}{4}} A = \frac{1}{2} [\frac{π}{4} + \frac{\sin 4 \frac{π}{4}}{4} - 0] A = \frac{π}{8} {unit}^{2}$

The area is $A = \frac{π}{8} {unit}^{2}$

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Areas of regions bounded by polar functions: Find the areas of the following regions. The area bounded by one petal of $r = \cos 2 θ$

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 one petal of r=cos2θ

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

Discrete Mathematics

Calculus

Mechanics Maths

Applied Mathematics

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Theoretical and Mathematical Physics

Study anywhere. Anytime. Across all devices.

Areas of regions bounded by polar functions: Find the areas of the following regions. The area bounded by one petal of $r = \cos 2 θ$