   Chapter 5.2, Problem 9E ### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

#### Solutions

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

8th Edition
James Stewart
ISBN: 9781305270343
Textbook Problem

# Use the Midpoint Rule with the given value of n to approximate the integral. Round the answer to four decimal places. ∫ 0 8 sin x   d x ,   n = 4

To determine

To evaluate: The integral usingmidpoint rule.

Explanation

Given information:

The integral function 08sinxdx and n=4.

Calculation:

Apply Midpoint Rule.

abf(x)dxi=1nf(x¯i)Δx=Δx[f(x¯1)+...+f(x¯n)] (1)

Find the width (Δx) using the relation:

Δx=ban

Substitute 8 for b, 0 for a and 4 for n in Equation (1).

Δx=804=84=2

Calculate right end points xi using the relation:

xi=a+iΔx (2)

Calculate left end points xi1 using Equation (2)

Substitute (i1) for i in Equation (2)

xi1=a+(i1)Δx (3)

Calculate mid points using the relation:

x¯i=12(xi1+xi)

Substitute (a+iΔx) for xi and a+(i1)Δx for xi1

x¯i=12(xi1+xi)=12[a+(i1)Δx+a+iΔx]=12(a+iΔxΔx+a+iΔx)=12×[2(a+iΔx)Δx]

x¯i=(a+iΔx)Δx2 (4)

Calculate x¯1 using Equation (4)

Substitute 0 for a, 1 for i,and 2 for Δx in Equation (4).

x¯i=(a+iΔx)Δx2=(0+1×2)22=21=1

Calculate x¯2 using Equation (4).

Substitute 0 for a, 2 for i,and 2 for Δx in Equation (4).

x¯i=(a+iΔx)Δx2=(0+2×2)22=41=3

Calculate x¯3 using Equation (4).

Substitute 0 for a, 3 for i,and 2 for Δx in Equation (4)

x¯i=(a+iΔx)Δx2=(0+3×2)22=61=5

Calculate x¯4 using Equation (4)

Substitute 0 for a, 3 for i,and 2 for Δx in Equation (4)

x¯i=(a+iΔx)Δx2=(0+4×2)22=81=7

Compare the integral function 08sinxdx with Equation (1)

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