# The integral usingmidpoint rule.

### Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805

### Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805

#### Solutions

Chapter 5.2, Problem 9E
To determine

## To evaluate: The integral usingmidpoint rule.

Expert Solution

The value of integral 08sinxdx is 6.1820_.

### Explanation of Solution

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).

f(x)=sinx

Calculate f(xi¯).

Substitute x¯i for x.

f(x¯i)=sinx¯i (5)

Calculate f(x¯1) using Equation (5).

Substitute 1 for x¯1 in the Equation (5)

f(x¯1)=sinx¯i=sin(1)=0.8415

Calculate f(x¯2) using Equation (5).

Substitute 3 for x¯2 in the Equation (5)

f(x¯2)=sinx¯2=sin(3)=0.9870

Calculate f(x¯3) using Equation (5).

Substitute 5 for x¯3 in the Equation (5).

f(x¯3)=sinx¯3=sin(5)=0.7867

Calculate f(x¯4) using the Equation (5).

Substitute 7 for x¯4 in the Equation (5).

f(x¯4)=sinx¯4=sin(7)=0.4758

Calculate 08sinxdx using Equation (1):

Substitute sinx for f(x), 2 for Δx,0.8415 for f(x¯1), 0.9870 for f(x¯2), 0.7867 for f(x¯3),n=4 and 0.4758 for f(x¯4), 0 for a and 8 for b in Equation (1).

08sinxdx=i=14f(x¯i)Δx=Δx[f(x¯1)+f(x¯2)+f(x¯3)+f(x¯4)]=2[0.8415+0.9870+0.7867+0.4758]=2×3.0910

08sinxdx=6.1820

Thus, the value of 08sinxdx is 6.1820.

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