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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

(a) Write 1 5 ( x + 2 x 5 ) d x as a limit of Riemann sums, taking the sample points to be right endpoints. Use a computer algebra system to evaluate the sum and to compute the limit.

(b) Use the Fundamental Theorem to check your answer to part (a).

(a)

To determine

The sum value of the function using computer algebra system.

Explanation

Given information:

The function as 15(x+2x5)dx.

The region lies between x=1 and x=5.

The expression to find the sum value when the sample points at right endpoints are shown below:

15(x+2x5)dx=limni=1nf(xi)Δx (1)

Here, the height of sample at the right endpoint is f(xi) and the width is Δx.

Find the width (Δx) using the relation:

Δx=ban (2)

Here, the upper limit is b, the lower limit is a, and the number of rectangles is n.

Substitute 5 for b and 1 for a in Equation (2).

Δx=51n=4n

To find the value of xi using the relation:

xi=a+iΔx (3)

Substitute 1 for a and 4n for Δx in Equation (3).

xi=1+i4n=1+4in

Substitute (xi+2xi5) for f(xi) in Equation (1).

15(x+2x5)dx=limni=1n(xi+2xi5)Δx (4)

Substitute 4n for Δx and 1+4in for xi in Equation (4)

(b)

To determine

The value of the integral using the fundamental theorem.

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