   Chapter 10.4, Problem 51E

Chapter
Section
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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