# The lower estimate and upper estimate for the integral ∫ 10 30 f ( x ) d x . ### 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 7E
To determine

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

Expert Solution

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_.

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