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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) Evaluate the Riemann sum for

f ( x ) = x 2 x 0 x 2

with four subintervals, taking the sample points to be right endpoints. Explain, with the aid of a diagram, what the Riemann sum represents.

(b) Use the definition of a definite integral (with right endpoints) to calculate the value of the integral

0 2 ( x 2 x ) d x

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

(d) Draw a diagram to explain the geometric meaning of the integral in part (b).

(a)

To determine

The Riemann sum with four subintervals for the function at the right endpoints.

Explanation

Given information:

The function as f(x)=x2x

The region lies between x=0 and x=2. Hence, the limits are a=0 and b=2.

Consider that the Number of rectangles as n=4.

The expression to find Riemann sum at the right endpoint (Rn) is shown below:

Rn=i=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 2 for b, 0 for a, and 4 for n in Equation (2).

Δx=204=0.5

Calculate R4 using equation (1).

Substitute 4 for n in equation (1).

R4=i=14f(xi)Δx (3)

Substitute i=1,2,3,and 4 in equation (3).

R4=f(x1)Δx+f(x2)Δx+f(x3)Δx+f(x4)Δx (4)

At x1=0

(b)

To determine

The value of the integral using the definition of a definite integral.

(c)

To determine

The value of the integral using the fundamental theorem.

(d)

To determine

Draw a diagram to explain the geometric meaning of the integral.

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