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Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270336

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BuyFindarrow_forward

Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270336
Textbook Problem

Use a calculator to find the length of the curve correct to four decimal places. If necessary, graph the curve to determine the parameter interval.

51. One loop of the curve r = cos 2θ

To determine

To Find: The length of the curve.

Explanation

Given:

One loop of the curve is as below.

r=cos2θ (1)

Calculation:

Assume the value of θ=0 .

Calculate the value of r from the equation (1).

r=cos2θ

Substitute 0 for θ .

r=cos2(0)=1

Calculate the value of x.

x=rcosθ

Substitute 1 for r and 0 for θ .

x=rcosθ=1×cos(0×π180)=1

Calculate the value of y.

y=rsinθ

Substitute 1 for r and 0 for θ .

y=1×sin(0×π180)=0

Repeat the calculation of the values of x and y using the value of θ from 0 to 2π .

Tabulate the values of x and y in table (1).

θ r=cos2θ x=rcosθ y=rsinθ
0.00 1.00 1.00 0.00
10.00 0.94 0.93 0.16
20.00 0.77 0.72 0.26
30.00 0.50 0.43 0.25
40.00 0.17 0.13 0.11
50.00 -0.17 -0.11 -0.13
60.00 -0.50 -0.25 -0.43
70.00 -0.77 -0.26 -0.72
80.00 -0.94 -0.16 -0.93
90.00 -1.00 0.00 -1.00
100.00 -0.94 0.16 -0.93
110.00 -0.77 0.26 -0.72
120.00 -0.50 0.25 -0.43
130.00 -0.17 0.11 -0.13
140.00 0.17 -0.13 0.11
150.00 0.50 -0.43 0.25
160.00 0.77 -0.72 0.26
170.00 0.94 -0.93 0.16
180.00 1.00 -1.00 0.00
190.00 0.94 -0.93 -0.16
200.00 0.77 -0.72 -0.26
210.00 0.50 -0.43 -0.25
220.00 0.17 -0.13 -0.11
230.00 -0.17 0.11 0.13
240.00 -0.50 0.25 0.43
250.00 -0.77 0.26 0.72
260.00 -0.94 0.16 0.93
270.00 -1.00 0.00 1.00
280.00 -0.94 -0.16 0.93
290.00 -0.77 -0.26 0.72
300.00 -0.50 -0.25 0.43
310.00 -0.17 -0.11 0.13
320.00 0.17 0.13 -0.11
330.00 0.50 0.43 -0.25
340.00 0.77 0.72 -0.26
350.00 0.94 0.93 -0.16
360.00 1.00 1.00 0.00

Graph:

The graph is plotted between x and y values as shown in figure (1).

Refer the figure (1).

The upper half of the right loop is completed from range of values in interval [0,π4]

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