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 8E
To determine

To estimate: The integral 39f(x)dx using three equal subintervals with right end points, left end points and midpoints and compare it with the exact value of integral.

Expert Solution

Answer to Problem 8E

The value of the integral 39f(x)dx for the right end points is 4.2_.

The value of the integral 39f(x)dx for the left end points is 6.2_.

The value of the integral 39f(x)dx for the mid points is 0.8_.

Explanation of Solution

Given information:

The function f is an increasing function.

The number of intervals, n=3.

Each interval value is Δx=2.

The Riemann sum Rn provides the right end point, L5 provides the left end points, and Mn provides the mid points estimate for the integral 39f(x)dx.

The expression to find the integral 39f(x)dx with right end points is shown below:

Rn=i=13f(xi)Δx=Δx[f(x9)+f(x7)+f(x5)] (1)

Substitute3 for n, 2 for Δx, and corresponding x values in Equation (1).

R3=2[f(9)+f(7)+f(5)] (2)

Substitute 1.8 for f(9), 0.9 for f(7), and 0.6 for f(5) in Equation (2).

R3=2[f(9)+f(7)+f(5)]=2(1.8+0.90.6)=4.2

Therefore, the estimation of the integral 39f(x)dx with right end points is 4.2_.

For the increasing function of f,the right end points are greater than the exact value of the integral that is overestimate.

The expression to find the integral 39f(x)dx with left end points is shown below:

Rn=i=13f(xi1)Δx=Δx[f(x3)+f(x5)+f(x7)] (3)

Substitute 3 for n,2 for Δx, and corresponding x values in Equation (3).

L3=2[f(3)+f(5)+f(7)] (4)

Substitute 3.4 for f(3), 0.6 for f(5), and 0.9 for f(7) in Equation (4).

L3=2[f(3)+f(5)+f(7)]=2(3.40.6+0.9)=6.2

Therefore, the estimation of the integral 39f(x)dx with left end points is 6.2_.

For the increasing function of f,the left end points are smaller than the exact value of the integral that is underestimating.

The expression to find the integral 39f(x)dx with mid points is shown below:

Mn=i=13f(x¯i)Δx=Δx[f(x¯4)+f(x¯6)+f(x¯8)] (5)

Substitute 3 for n,2 for Δx, and corresponding x values in Equation (5).

M3=2[f(4)+f(6)+f(8)] (6)

Substitute 2.1 for f(4), 0.3 for f(6), and 1.4 for f(8) in Equation (6).

M3=2[f(4)+f(6)+f(8)]=2(2.1+0.3+1.4)=0.8

Therefore, the estimation of the integral 39f(x)dx with mid points is 0.8_.

Due to increasing function of f, cannot be concluded about midpoint estimate when compared to the exact value of integral.

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