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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 graph of g is shown. Estimate 2 4 g ( x ) d x with six subintervals using (a) right endpoints, (b) left endpoints, and (c) midpoints.

Chapter 5.2, Problem 6E, The graph of g is shown. Estimate 24g(x)dx with six subintervals using (a) right endpoints, (b) left

(a)

To determine

To estimate: The value of the integral 24g(x)dx by using the right end points.

Explanation

The expression of the Riemann sum using the right end points is shown below:

Rn=i=1nf(xi)Δx=Δx(f(x1)++f(xn)) (1)

Apply the condition for Riemann sum as follows:

Δx=ban (2)

Here, width of the subinterval is Δx, the upper limit is b, the lower limit is a, the number of subinterval is n, and the right endpoint of (xi1,xi) is xi1.

Find the width of the subinterval (Δx) using Equation (2) as shown below.

Consider the number of subintervals as 6.

Substitute 6 for n, 4 for b and -2 for a in Equation (2).

Δx=4(2)6=1

Calculate the value of Riemann sum using the right end points.

Refer to the graph of the function g.

The right end points of the intervals are -1, 0, 1, 2, 3, and 4.

Substitute 6 for n, 1 for Δx, -1 for x1, 0 for x2, 1 for x3, 2 for x4, 3 for x5, and 4 for x6 in Equation (1) as shown below

(b)

To determine

To estimate: The value of the integral 24g(x)dx using the left end points.

(c)

To determine

To estimate: The value of the integral 24g(x)dx by using the midpoints.

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