   Chapter 8.1, Problem 26E

Chapter
Section
Textbook Problem

Use Simpson’s Rule with n = 10 to estimate the arc length of the curve. Compare your answer with the value of the integral produced by a calculator.26. y = x 3 , 1 ≤ x ≤ 6

To determine

To find: The arc length of the curve.

Explanation

Given information:

The curve function is y=x3 (1)

The lower limit is a=1 and the upper limit is b=6.

The number of intervals is 10.

Calculation:

The Simpson’s rule is shown below:

abf(x)dxSn=Δx3[f(x0)+4f(x1)+2f(x2)+4f(x3)+...+2f(xn2)+4f(xn1)+f(xn)] (2)

The interval width Δx=ban (3)

Here, the value of the function f(x) at x=0ton is f(x0)tof(xn), the upper limit is b, the lower limit is a, and the number of interval is n.

Find the value of f(x) using the relation:

f(x)=1+(dydx)2 (4)

Here, the derivative of the function y is dydx.

Differentiate Equation (1) with respect to x.

dydx=13x23

Substitute 13x23 for dydx in Equation (4).

f(x)=1+(13x23)2=1+19x43 (5)

Find the width of the interval:

Substitute 6 for b, 1 for a, and 10 for n in Equation (3).

Δx=6110=12

Modify Simpson’s rule for n=10.

S10=Δx3[f(x0)+4f(x1)+2f(x2)+4f(x3)+...+2f(x8)+4f(x9)+f(x10)] (6)

Split the interval 1 to 6 into 10 subintervals. Thus the intervals are 1,1.5,2,2.5,3,3.5,4,4.5,5,5.5,and6.

Substitute 1 for x0, 1.5 for x1, 2 for x2, 2.5 for x3, 3 for x4, 3

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