   Chapter 4.R, Problem 2E

Chapter
Section
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).

To determine

a)

To evaluate:

The Riemann sum for fx

Explanation

Concept:

Definition of Riemann sum:

i=1nfxi* x

is called as Riemann sum

Where, x= b-an,    xi=a+i x,    &xi*xi-1, xi

Riemann sum is the sum, of the areas of the rectangles that lie above the x-axis and the negative of the areas of the rectangles that lie below the x-axis

Given:

fx=x2-x,  0x2

Calculation:

First, write the expression of the Riemann sum

Four subintervals are given, that means, n=4

Taking the sample points as the right endpoints means that

The right endpoints are 0.5, 1, 1.5, 2

By definition of Riemann sum

i=1nfxi x=i=14fxi x

So now calculate  x,

x= b-an=2-04=12

Substitute this value in Riemann sum expression,

=i=14fxi

To determine

b)

To calculate:

The value of the integral 02x2-xdx

To determine

c)

To check:

02x2-xdx=23

To determine

d)

To draw:

Graph of the function  fx=x2-x

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