   Chapter 8.2, Problem 22E

Chapter
Section
Textbook Problem

Use Simpson’s Rule with n = 10 to approximate the area of the surface obtained by rotating the curve about the x-axis. Compare your answer with the value of the integral produced by a calculator.22. y = x ln x, 1 ≤ x ≤ 2

To determine

To Calculate: the approximate value of the area of the surface obtained by rotating the curve about x-axis using Simpson’s rule.

To compare: the value of the integral produced by the calculator.

Explanation

Given information:

The Equation of the curve is y=xlnx,1x2 .

The number of sub intervals as n=10 .

Calculation:

Show the equation of the curve.

y=xlnx (1)

Calculate the area of the surface obtained by rotating the curve about x-axis using the relation:

S=ab2πy1+(dydx)2dx (2)

Here, S is the area of the surface obtained by rotating the curve about y-axis and

axb .

Differentiate both sides of Equation (1) with respect to x.

dydx=ddx(xlnx)

Show the formula of differentiation.

ddx(uv)=ududx+vdvdx

Apply the formula of differentiation.

dydx=ddx(xlnx)=xd(lnx)dx+lnxdxdx=x(1x)+lnx=1+lnx

Substitute xlnx for y, (1+lnx) for dydx , 1 for a, and 2 for b in Equation (2).

S=012πxex1+(xex+ex)2dx (3)

Consider the function f(x)=2πxlnx1+(1+lnx)2 .

Calculate the length of the subinterval (Δx) using the formula:

Δx=ban

Here, b is upper limit, a is lower limit, and n is number of subintervals.

Substitute 2 for b, 1 for a, and 10 for n.

Δx=2110=110=0.1

Calculate the approximate value of integral by using Simpson’s rule.

Show the Simpson’s rule as follows:

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

Consider the condition for Simpson rule as follows:

• Δx=ban
• Number of subinterval n is even.

Here, Sn is Simpson’s approximation, x1,x2,...xn are subintervals, b is upper limit, a is lower limit, and n is number of subintervals.

Apply Simpson’s rule in Equation (3).

012πxlnx1+(1+lnx)2dxSn=Δx3[f(x0)+4f(x1)+2f(x2)+4f(x3)...+2f(xn2)+4f(xn1)+f(xn)]

Substitute 0

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