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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 $r = a \cos 胃$ , where $a$ is a positive constant.

Short Answer

Expert verified

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

Step by step solution

01

Given information.

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

02

Area

The area of the curve is given by,

$A = \frac{1}{2} {鈭�}_{伪}^{尾} r^{2} d 胃 A = \frac{1}{2} {鈭�}_{0}^{2 蟺} {(a \cos 胃)}^{2} d 胃 A = \frac{1}{2} {鈭�}_{0}^{2 蟺} (a^{2} \cos^{2} 胃) d 胃 A = \frac{a^{2}}{2} {鈭�}_{0}^{2 蟺} (\frac{1 + \cos 2 胃}{2}) d 胃 A = \frac{a^{2}}{4} {鈭�}_{0}^{2 蟺} (\frac{1 + \cos 2 胃}{2}) d 胃 A = \frac{a^{2}}{4} {[胃 + \frac{\sin 2 胃}{2}]}_{0}^{2 蟺} A A = \frac{a^{2}}{4} {[2 蟺 + \frac{\sin 4 蟺}{2}]}_{0}^{2 蟺} A = \frac{{蟺补}^{2}}{2} {unit}^{2}$

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

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Most popular questions from this chapter

In Exercises 32鈥�47 convert the equations given in polar coordinates to rectangular coordinates.

$r = - 3 \sec 胃$

In Exercises 32鈥�47 convert the equations given in polar coordinates to rectangular coordinates.

$r^{2} = \cos 胃$

in exercise 31-36 find a definite integral that represents the length of the specified polar curve, and then use graphing calculator or computer algebra system to approximate the value of integral

The limacon $r = 2 + \cos 胃$

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

In exercise 26-30 Find a definite integral that represents the length of the specified polar curve and then find the exact value of integral

$r = e^{胃} f o r 2 k 蟺鈮� 胃鈮� 2 (k + 1) 蟺$

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