   Chapter 5.2, Problem 10E ### 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 1 x 3 + 1   d x ,   n = 5

To determine

To evaluate: The integral using Midpoint Rule.

Explanation

Given information:

The integral function 01(x3+1)dx,n=5.

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 1 for b, 0 for a, and 5 for n in Equation (1).

Δx=105=15=0.2

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 0.2 for Δx in Equation (4).

x¯i=(a+iΔx)Δx2=(0+1×0.2)12×(0.2)=0.20.1=0.1

Calculate x¯2 using Equation (4).

Substitute 0 for a, 2 for i,and 0.2 for Δx in Equation (4).

x¯i=(a+iΔx)Δx2=(0+2×0.2)0.22=0.40.1=0.3

Calculate x¯3 using Equation (4)

Substitute 0 for a, 3 for i,and 0.2 for Δx in Equation (4)

x¯i=(a+iΔx)Δx2=(0+3×0.2)0.22=0.60.1=0.5

Calculate x¯4 using Equation (4)

Substitute 0 for a, 4 for i,and 0.2 for Δx in Equation (4)

x¯i=(a+iΔx)Δx2=(0+4×0.2)0.22=0.80.1=0.7

Calculate x¯5 using Equation (4)

Substitute 0 for a, 5 for i,and 0.2 for Δx in Equation (4)

x¯i=(a+iΔx)Δx2=(0+5×0.2)0.22=10.1=0.9

Compare the integral function 01(x3+1)dx with Equation (1)

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