   Chapter 10.4, Problem 50E

Chapter
Section
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

Still sussing out bartleby?

Check out a sample textbook solution.

See a sample solution

The Solution to Your Study Problems

Bartleby provides explanations to thousands of textbook problems written by our experts, many with advanced degrees!

Get Started

Draw the graph of 3x2y=12.

Elementary Technical Mathematics

In Exercises 21 to 32, solve each equation. y3+2=6

Elementary Geometry For College Students, 7e

2. If , form , the average cost function, and find the minimum average cost.

Mathematical Applications for the Management, Life, and Social Sciences

In Exercises 57-62, Use Venn diagram to illustrate each statement. ABA;ABB

Finite Mathematics for the Managerial, Life, and Social Sciences

True or False: If , then converge absolutely.

Study Guide for Stewart's Multivariable Calculus, 8th

True or False: The figure at the right is the graph of the direction field for y = (y 1)x2.

Study Guide for Stewart's Single Variable Calculus: Early Transcendentals, 8th 