   Chapter 5.2, Problem 1E ### 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

# Evaluate the Riemann sum for f(x) = x − 1, − 6 ≤ x ≤ 4, with five subintervals, taking the sample points to be right endpoints. Explain, with the aid of a diagram, what the Riemann sum represents.

To determine

To evaluate: The Riemann sum for the function f(x)=x1,6x4 with five subintervals and interpret the result using diagram.

Explanation

Take the function as f(x)=x1,6x4.

The region lies between x=6 and x=4. So the limits are a=6 and b=4.

Consider the number of rectangles as n=5.

Find the width (Δx) using the relation:

Δx=ban (1)

Here, the upper limit is b, the lower limit is a, and the number of rectangles is n.

Substitute 4 for b, -6 for a and 5 for n in Equation (1).

Δx=4(6)5=105=2

Determine right end points using the relation:

xi=a+iΔx (2)

Calculate x1 using equation (2).

Substitute 1 for i and 2 for Δx in equation (2).

x1=6+(1×2)=4

Calculate x2 using equation (2).

Substitute 2 for i and 2 for Δx in Equation (2).

x2=6+(2×2)=2

Calculate x3 using equation (2).

Substitute 3 for i and 2 for Δx in Equation (2).

x3=6+(3×2)=0

Calculate x4 using equation (2).

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

x4=6+(4×2)=2

Calculate x5 using equation (2)

Substitute 5 for i and 2 for Δx in Equation (2).

x5=6+(5×2)=4

Determine f(xi) using the equation:

f(xi)=xi1 (3)

Calculate f(x1) using equation (3).

Substitute 1 for i and -4 for x1 in the equation (3).

f(x1)=x11=41=5

Calculate f(x2) using equation (3).

Substitute 2 for i and -2 for x2 in the equation (3).

f(x2)=x21=21=3

Calculate f(x3) using equation (3).

Substitute 3 for i and 0 for x3 in the equation (3).

f(x3)=x31=01=1

Calculate f(x4) using equation (3)

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