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Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

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Section
BuyFindarrow_forward

Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343
Textbook Problem

The table gives the values of a function obtained from an experiment. Use them to estimate 3 9 f ( x ) d x using three equal subintervals with (a) right endpoints, (b) left endpoints, and (c) midpoints. If the function is known to be an increasing function, can you say whether your estimates are less than or greater than the exact value of the integral?

Chapter 5.2, Problem 8E, The table gives the values of a function obtained from an experiment. Use them to estimate 39f(x)dx

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.

Explanation

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

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