Chapter 9: Q. 34 (page 775)

Areas of regions bounded by polar functions: Find the areas of the following regions. The area bounded by one petal of $r = \sin 3 θ$

Short Answer

Expert verified

The area bounded by one petal is $\frac{π}{6} {unit}^{2}$

Step by step solution

Given information

The curve is $r = \sin 3 θ$

Area

The area bounded by the curve is,

$A = \frac{1}{2} \int_{α}^{β} r^{2} d θ A = \frac{1}{2} \int_{0}^{\frac{π}{3}} {(\sin 3 θ)}^{2} d θ A = \frac{1}{2} \int_{0}^{\frac{π}{3}} (\sin^{2} 3 θ) d θ A = \frac{1}{2} \int_{0}^{\frac{π}{3}} \frac{(1 - \cos 6 θ)}{2} d θ A = \frac{1}{2} {[θ - \frac{\sin 6 θ}{6}]}_{0}^{\frac{π}{3}} A = \frac{1}{2} [\frac{π}{3} - \frac{\sin 6 \frac{π}{3}}{6} - 0] A = \frac{π}{6} {unit}^{2}$

The area of one petal is $A = \frac{π}{6} {unit}^{2}$

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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 $r = \sin 3 θ$

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=sin3θ

Short Answer

Step by step solution

Given information

Area

One App. One Place for Learning.

Most popular questions from this chapter

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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 $r = \sin 3 θ$