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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 $r = \sin 3 胃$

Short Answer

Expert verified

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

Step by step solution

01

Given information

The curve is $r = \sin 3 胃$

02

Area

The area bounded by the curve is,

$A = \frac{1}{2} {鈭�}_{伪}^{尾} r^{2} d 胃 A = \frac{1}{2} {鈭�}_{0}^{\frac{蟺}{3}} {(\sin 3 胃)}^{2} d 胃 A = \frac{1}{2} {鈭�}_{0}^{\frac{蟺}{3}} (\sin^{2} 3 胃) d 胃 A = \frac{1}{2} {鈭�}_{0}^{\frac{蟺}{3}} \frac{(1 - \cos 6 胃)}{2} d 胃 A = \frac{1}{2} {[胃 - \frac{\sin 6 胃}{6}]}_{0}^{\frac{蟺}{3}} A = \frac{1}{2} [\frac{蟺}{3} - \frac{\sin 6 \frac{蟺}{3}}{6} - 0] A = \frac{蟺}{6} {unit}^{2}$

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

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Most popular questions from this chapter

Use Cartesian coordinates to express the equations for the parabolas determined by the conditions specified in Exercises 22鈥�31.

$r = \frac{伪}{1 + \sin 胃}$

Explain why there are infinitely many different hyperbolas with the same foci.

Sketch the graphs of the equations

$r = \frac{2}{2 + \cos 胃}$ and localid="1649860998050" $r = \frac{2}{2 + \sin 胃}$

What is the relationship between these graphs? What is the eccentricity of each graph?

In Exercises 24鈥�31 find all polar coordinate representations for the point given in rectangular coordinates.

$(0, 2)$

Use Cartesian coordinates to express the equations for the ellipses determined by the conditions specified in Exercises 32鈥�37.

$foci (0, 卤伪), minor axis 2 伪$

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