   Chapter 8.1, Problem 46E

Chapter
Section
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,n=4,6,8 (1)

Draw the curve for the function xn+yn=1 with n=2,4,6,8and10 as shown in Figure 1.

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

L=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=1x2ky=(1x2k)12k (4)

Differentiate Equation (4) with respect to x.

dydx=12k(1x2k)12k1(2kx2k1)=x2k1(1x2k)12k1

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=4011+(dydx)2dx (5)

Substitute x2k1(1x2k)12k1 for dydx in Equation (5)

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