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Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

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BuyFindarrow_forward

Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343
Textbook Problem

If f(x) = x2 − 4, 0 ≤ x ≤ 3, find the Riemann sum with n = 6, taking the sample points to be midpoints. What does the Riemann sum represent? Illustrate with a diagram.

To determine

The Riemann sum for the function f(x)=x24 and explain the meaning of it.

Explanation

Given:

The function is f(x)=x24.

The region lies between x=0 and x=3.

Number of rectangles is n=6.

Calculation:

Find the width (Δx) using the relation:

(Δx)=ban (1)

Substitute 2 for b, 0 for a, and 5 for n in Equation (1).

Δx=306=12

Calculate the value of xi using relation:

xi=a+iΔx (2)

Calculate the value of xi+1 using Equation (2)

Substitute (i+1) for i in Equation (2)

xi1=a+(i1)Δx (3)

Calculate the value of mid points using the relation:

xi¯=12(xi1+xi)

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

xi¯=12[a+(i1)Δx+a+iΔx]=12(a+iΔxΔx+a+iΔx)=12[2(a+iΔx)Δx]

=(a+iΔx)Δx2 (4)

Calculate the value of xi¯ for six subintervals:

Substitute 0 for a, 1 for i and 12 for Δx in Equation (4).

x1¯=(0+1×12)12(12)=1214=14

Substitute 0 for a, 2 for i, and 12 for Δx in Equation (4).

x2¯=(0+2×12)12(12)=114=34

Substitute 0 for a, 3 for i, and 12 for Δx in Equation (4).

x3¯=(0+3×12)12(12)=3214=54

Substitute 0 for a, 4 for i, and 12 for Δx in Equation (4).

x4¯=(0+4×12)12(12)=214=74

Substitute 0 for a, 5 for i, and 12 for Δx in Equation (4).

x5¯=(0+5×12)12(12)=5214=94

Substitute 0 for a, 6 for i, and 12 for Δx in Equation (4).

x6¯=(0+6×12)12(12)=314=114

Substitute xi¯ for x in the function f(x)=x24.

f(xi¯)=(xi¯)24 (5)

Calculate the value of f(xi¯) using Equation (5).

Substitute 14 for x1¯ in Equation (5).

f(x1¯)=(14)24=1164=6316

Substitute 34 for x2¯ in Equation (5).

f(x2¯)=(34)24=9164=5516

Substitute 54 for x3¯ in Equation (5)

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