menu
Hit Return to see all results
Chapter 8.1, Problem 46E
BuyFindarrow_forward

### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270336

#### Solutions

Chapter
Section
BuyFindarrow_forward

### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270336
Textbook Problem

# The curves with equations x″ + y″ = l , n = 4, 6, 8, …, are called fat circles. Graph the curves with n = 2, 4, 6, 8, and 10 to see why. Set up an integral for the length L2k of the fat circle with n = 2k. Without attempting to evaluate this integral, state the value of lim k→∞ L2k.

To determine

To find: An integral for the length L2k and evaluate the value of limkL2k.

Explanation

The curve function is xn+yn=1,â€‰â€‰n=4,â€‰6,â€‰8 (1)

Draw the curve for the function xn+yn=1 with n=2,â€‰4,â€‰6,â€‰8â€‰andâ€‰10 as shown in Figure 1.

The expression to find the length of the curve (L) is shown below:

L=âˆ«ab1+(dydx)2dx (2)

Here, the derivative of the function y is dydx, the lower limit is a, and the upper limit is b.

Substitute 2k for n in Equation (1).

x2k+y2k=1 (3)

Rearrange Equation (3) as shown below.

y2k=1âˆ’x2ky=(1âˆ’x2k)12k (4)

Differentiate Equation (4) with respect to x.

dydx=12k(1âˆ’x2k)12kâˆ’1(âˆ’2kx2kâˆ’1)=âˆ’x2kâˆ’1(1âˆ’x2k)12kâˆ’1

Refer Figure 1.

The curves are symmetric.

The total length is equal to 4 times the length of the first quadrant.

The limits of the first quadrant is x=0 to x=1.

Modify Equation (2) using the limits of first quadrant and n=2k.

L2k=4âˆ«011+(dydx)2dx (5)

Substitute âˆ’x2kâˆ’1(1âˆ’x2k)12kâˆ’1 for dydx in Equation (5)

### 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

## Additional Math Solutions

#### Find more solutions based on key concepts

Show solutions add

#### In Exercises 107-120, factor each expression completely. 107. 6ac + 3bc 4ad 2bd

Applied Calculus for the Managerial, Life, and Social Sciences: A Brief Approach

#### In Exercises 5-8, solve for x. 3x2+8x+4=0

Calculus: An Applied Approach (MindTap Course List)