   Chapter 8, Problem 8RE

Chapter
Section
Textbook Problem

Use Simpson’s Rule with n = 10 to estimate the area of the surface obtained by rotating the sine curve in Exercise 7 about the x-axis.

To determine

To find: The area of the surface obtained by rotating the sine curve about x-axis using Simpson’s rule.

Explanation

Given information:

The curve function is y=sinx (1)

The lower limit is a=0 and the upper limit is b=π .

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.

The expression to find the area of the surface (S) obtained by rotating the curve about the x-axis is shown below:

S=ab2πy1+(dydx)2dx (4)

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

Here, the derivative of the function y is dydx .

Differentiate Equation (1) with respect to x.

dydx=cosx

Substitute sinx for y and cosx for dydx in Equation (4).

S=ab2πsinx1+cos2xdx (5)

Let f(x)=2πsinx1+cos2x (6)

Find the width of the interval using Equation (3):

Substitute π for b, 0 for a, and 10 for n in Equation (3).

Δx=π010=π10

Modify Simpson’s rule for n=10 .

S10=Δx3[f(x0)+4f(x1)+2f(x2)+4f(x3)+...+2f(x8)+4f(x9)+f(x10)] (7)

Split the interval 0 to π into 10 subintervals. Thus the intervals are 0,π10,2π10,3π10,4π10,5π10,6π10,7π10,8π10,9π10,andπ

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