Chapter 9: Q. 40 (page 775)

Areas of regions bounded by polar functions: Find the areas of the following regions. The area inside both polar roses $r = \sin 3 θ and r = \cos 3 θ$ .

Short Answer

Expert verified

The area inside both polar roses is $(\frac{1}{6} - \frac{\sqrt{2 - \sqrt{2}}}{6}) units$

Step by step solution

Given information

$r = \sin 3 θ and r = \cos 3 θ$

Area

The graph of both the curves is,

The limits of the integration is,

$\sin 3 θ = \cos 3 θ \Rightarrow \tan 3 θ = 1 \Rightarrow 3 θ = \frac{π}{4} \Rightarrow θ = \frac{π}{12}$

The area bounded by both the curves is,

$A = \frac{1}{2} \int_{α}^{β} r^{2} d θ A = 3 \times \frac{1}{2} \int_{0}^{\frac{π}{12}} \sin 3 θ d θ A = \frac{1}{2} {(- \frac{\cos 3 θ}{3})}_{0}^{\frac{π}{8}} A = \frac{1}{2} (- \frac{\cos 3 (\frac{π}{8})}{3} + \frac{\cos 3 (0)}{3}) A = (\frac{1}{6} - \frac{\sqrt{2 - \sqrt{2}}}{6}) units$

The required area is $A = (\frac{1}{6} - \frac{\sqrt{2 - \sqrt{2}}}{6}) units$

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Areas of regions bounded by polar functions: Find the areas of the following regions. The area inside both polar roses $r = \sin 3 θ and r = \cos 3 θ$ .

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 inside both polar roses r=sin3θandr=cos3θ.

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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Study anywhere. Anytime. Across all devices.

Areas of regions bounded by polar functions: Find the areas of the following regions. The area inside both polar roses $r = \sin 3 θ and r = \cos 3 θ$ .