Chapter 8.2, Problem 22E

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

8th Edition
James Stewart
ISBN: 9781305270336

Chapter
Section

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

8th Edition
James Stewart
ISBN: 9781305270336
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,1â‰¤xâ‰¤2 .

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=âˆ«ab2Ï€y1+(dydx)2dx (2)

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

aâ‰¤xâ‰¤b .

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=âˆ«012Ï€xex1+(xex+ex)2dx (3)

Consider the function f(x)=2Ï€xlnx1+(1+lnx)2 .

Calculate the length of the subinterval (Î”x) using the formula:

Î”x=bâˆ’an

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.

Î”x=2âˆ’110=110=0.1

Calculate the approximate value of integral by using Simpsonâ€™s rule.

Show the Simpsonâ€™s rule as follows:

âˆ«abf(x)â€‰dxâ‰ˆSn=Î”x3[f(x0)+4f(x1)+2f(x2)+4f(x3)...+2f(xnâˆ’2)+4f(xnâˆ’1)+f(xn)]

Consider the condition for Simpson rule as follows:

• Î”x=bâˆ’an
• Number of subinterval n is even.

Here, Sn is Simpsonâ€™s approximation, x1,x2,â€‰...â€‰xn are subintervals, b is upper limit, a is lower limit, and n is number of subintervals.

Apply Simpsonâ€™s rule in Equation (3).

âˆ«012Ï€xlnx1+(1+lnx)2dxâ‰ˆSn=Î”x3[f(x0)+4f(x1)+2f(x2)+4f(x3)...+2f(xnâˆ’2)+4f(xnâˆ’1)+f(xn)]

Substitute 0

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