BuyFind

Single Variable Calculus: Concepts...

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

Single Variable Calculus: Concepts...

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

Solutions

Chapter 5.2, Problem 7E
To determine

The lower estimate and upper estimate for the integral 1030f(x)dx.

Expert Solution

Answer to Problem 7E

The lower estimate for the integral 1030f(x)dx is 64_.

The upper estimate for the integral 1030f(x)dx is 16_.

Explanation of Solution

Given information:

The function f is an increasing function.

The number of intervals is n=5.

The sub interval value is Δx=4.

Calculation:

Since the function f is increasing, the Riemann sum L5 provides the lower estimate and R5 provides the upper estimate for the integral 1030f(x)dx.

The expression to find the lower estimate is shown below:

Ln=i=15f(xi1)Δx=Δx[f(x0)+f(x1)+f(x2)+f(x3)+f(x4)] (1)

Substitute 5 for n, 4 for Δx, and corresponding x values in Equation (1).

L5=4[f(10)+f(14)+f(18)+f(22)+f(26)] (2)

Substitute 12 for f(10), 6 for f(14), 2 for f(18), 1 for f(22), and 3 for f(26) in Equation (2).

L5=4[f(10)+f(14)+f(18)+f(22)+f(26)]=4(1262+1+3)=64

Therefore, the lower estimate for the integral 1030f(x)dx is 64_.

The expression to find the upper estimate is shown below:

Rn=i=15f(xi)Δx=Δx[f(x1)+f(x2)+f(x3)+f(x4)+f(x5)] (3)

Substitute 5 for n, 4 for Δx, and corresponding x values in Equation (1).

R5=4[f(14)+f(18)+f(22)+f(26)+f(30)] (4)

Substitute 6 for f(14), 2 for f(18), 1 for f(22), 3 for f(26), and 8 for f(30),in Equation (4).

R5=4[f(14)+f(18)+f(22)+f(26)+f(30)]=4(62+1+3+8)=16

Therefore, the upper estimate for the integral 1030f(x)dx is 16_.

Have a homework question?

Subscribe to bartleby learn! Ask subject matter experts 30 homework questions each month. Plus, you’ll have access to millions of step-by-step textbook answers!