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Chapter 9: Q. 17. (page 756)

In Exercises 17鈥�25 find a definite integral expression that represents
the area of the given region in the polar plane, and
then find the exact value of the expression.
The region enclosed by the spiral $r = 胃$ and the $x$ -axis on
the interval $0 鈮� 胃鈮� 蟺$

Short Answer

Expert verified

The area of the spiral $r = 胃$ is $\frac{蟺^{3}}{6}$

Step by step solution

01

Given information

The spiral $r = 胃$ on the interval $0 鈮� 胃鈮� 蟺$

02

The objective is to find the area of the spiral.

The region's corresponding limits are $0 t o 蟺$

The interval is $[0, 蟺]$

The formula of the area is $A = {鈭�}_{伪}^{尾} \frac{1}{2} (f (胃))^{2} d 胃$ or $A = {鈭�}_{伪}^{尾} \frac{1}{2} r^{2} d 胃$

03

The area of the function is calculated as below

$A = \frac{1}{2} {鈭�}_{0}^{蟺} 胃^{2} d 胃 [since r = 胃] A = \frac{1}{2} {[\frac{胃^{3}}{3}]}_{0}^{蟺}$

Limits are established by applying them

$A = \frac{1}{2} [\frac{蟺^{3}}{3} - 0]$

$A = \frac{1}{2} 路 \frac{蟺^{3}}{3} 鈬� A = \frac{蟺^{3}}{6}$

The spiral's enclosing area $r = 胃$ is $A = \frac{蟺^{3}}{6}$

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Most popular questions from this chapter

Find a definite integral expression that represents the area of the given region in polar plane and then find the exact value of the expression

The region inside both of the cardioids $r = 3 + 3 \sin 胃 a n d r = 1 + \sin 胃$

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

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

$r^{2} = \cos 胃$

In Exercises 60 and 61 we ask you to prove Theorem 9.23 for ellipses and hyperbolas

Consider the ellipse with equation $\frac{x^{2}}{A^{2}} + \frac{y^{2}}{B^{2}} = 1$ where $A > B$ . Let $F$ be the focus with coordinates $(\sqrt{A^{2} - B^{2}}, 0)$ . Let $e = \frac{\sqrt{A^{2} - B^{2}}}{A}$ and l be the vertical line with equation $x = \frac{A}{e}$ . Show that for any point P on the ellipse, $\frac{F P}{D P} = e_{'}$ , where $D$ is the point on $l$ closest to $P$ .

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

$(- 2, 0)$

See all solutions

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