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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

Find the exact length of the curve. Use a graph to determine the parameter interval.

50. r = cos2(θ/2)

To determine

To find: The exact length of the polar curve.

Explanation

Given:

The polar equation is as below.

r=cos2(θ2) (1)

Calculation:

Assume the value of θ=0°.

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

r=cos2(θ2)

Substitute the value of 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=cos2(θ2) x=rcosθ y=rsinθ
0.00 1.00 1.00 0.00
10.00 0.99 0.98 0.17
20.00 0.97 0.91 0.33
30.00 0.93 0.81 0.47
40.00 0.88 0.68 0.57
50.00 0.82 0.53 0.63
60.00 0.75 0.38 0.65
70.00 0.67 0.23 0.63
80.00 0.59 0.10 0.58
90.00 0.50 0.00 0.50
100.00 0.41 −0.07 0.41
110.00 0.33 −0.11 0.31
120.00 0.25 −0.13 0.22
130.00 0.18 −0.11 0.14
140.00 0.12 −0.09 0.08
150.00 0.07 −0.06 0.03
160.00 0.03 −0.03 0.01
170.00 0.01 −0.01 0.00
180.00 0.00 0.00 0.00
190.00 0.01 −0.01 0.00
200.00 0.03 −0.03 −0.01
210.00 0.07 −0.06 −0.03
220.00 0.12 −0.09 −0.08
230.00 0.18 −0.11 −0.14
240.00 0.25 −0.13 −0.22
250.00 0.33 −0.11 −0.31
260.00 0.41 −0.07 −0.41
270.00 0.50 0.00 −0.50
280.00 0.59 0.10 −0.58
290.00 0.67 0.23 −0.63
300.00 0.75 0.38 −0.65
310.00 0.82 0.53 −0.63
320.00 0.88 0.68 −0.57
330.00 0.93 0.81 −0.47
340.00 0.97 0.91 −0.33
350.00 0.99 0.98 −0.17
360.00 1.00 1.00 0.00

Graph:

The graph is plotted for x and y as shown in Figure (1).

Refer the figure (1).

The complete figure is formed between the limits for θ from 0θ2π.

Use the symmetry and double the area of integral from 0 to π.

Calculate the value of drdθ.

Differentiate equation (1) with respect to θ.

Apply the chain rule.

df(u)dθ=dfdu.dudθ

Substitute u for cos(θ2) and u2 for f.

df(u)dθ=d(u2)du

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