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Chapter 9: Q. 8 (page 774)

Fill in the blanks to complete each of the following theorem statements:

Let r=f胃be a differentiable function of 胃such that f'(胃)is continuous for all 胃鈭�伪,尾Furthermore, assume that r=f(胃)is a one-to-one function from 伪,尾to the graph of the function. Then the length of the polar graph of r=f(胃)on the interval 伪,尾is..............

Short Answer

Expert verified

The blank is filled by鈭�伪尾f'胃2+f胃2d胃

Step by step solution

01

Step 1. Given information

r=f胃

伪鈮�胃鈮�尾

02

Step 2. Write formula of length of the arc.

As we know that when x and y are functions of the parameter 胃, the arc length of the curve on the interval [伪, 尾] is

l=鈭�伪尾dxd胃2+dyd胃2d胃

By combining this integral with the parametric equations for the curve,

x=rcos胃x=f胃cos胃and y=f胃sin胃

we have an arc length

l=鈭�伪尾dd胃f(胃cos胃)2+dd胃f(胃sin胃)2d胃=鈭�伪尾-f(胃)sin胃+f'(胃)cos胃2+f'(胃)sin胃+f(胃)cos胃2d胃

After expanding the terms under the radical and simplifying , we get:

l=鈭�伪尾f'胃2+f胃2d胃

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Most popular questions from this chapter

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

0,-3

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

r=伪1+cos胃

Three noncollinear points determine a unique circle. Do three noncollinear points determine a unique ellipse? If so, explain why. If not, provide three noncollinear points that are on two distinct ellipses.

Sketch the graphs of the equations

r=11+cos胃and localid="1649854142659" r=11+sin胃

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

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

r=5sin胃
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