   Chapter 5.2, Problem 71E ### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

#### Solutions

Chapter
Section ### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343
Textbook Problem

# Let f(x) = 0 if x is any rational number and f(x) = 1 if x is any irrational number. Show that f is not integrable on [0, 1].

To determine

To Show: The function f is not an integrable on [0,1].

Explanation

Given:

The value of the function f(x)=1, if x is irrational number.

The value of the function f(x)=0, if x is rational number.

Calculation:

Consider the function f is integrable within the limit [0,1].

Show the Theorem 4 as shown below:

If f is integrable on [a,b], then

abf(x)=limni=1nf(xi)Δx (1)

Refer to theorem 4.

The value of limni=1nf(xi)Δx exist as f is integrable on [0,1].

Consider a positive integer as n.

Divide the interval [0,1] into n equal interval.

Consider the xi* as a rational number in the ith sub interval.

The expression to find the Riemann sum Rn as shown below:

Rn=i=1nf(xi*)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 1 for b, 0 for a in Equation (2).

Δx=10n=1n

Calculate the Riemann sum using the Equation (1).

Substitute 1n for Δx in Equation (1).

Rn=i=1nf(xi*)1n

The value of f(xi*) is 0 as xi* is a rational number.

Rn=i=1n0×1n=0

Calculate the value of limni=1nf(xi*)Δx as shown below.

Substitute 0 for i=1nf(xi)Δx.

limni=1nf(xi*)Δx=limn(0)=0

Thus, the value of the limni=1nf(xi*)Δx is 0 for xi* to be rational number

### Still sussing out bartleby?

Check out a sample textbook solution.

See a sample solution

#### The Solution to Your Study Problems

Bartleby provides explanations to thousands of textbook problems written by our experts, many with advanced degrees!

Get Started

#### Find more solutions based on key concepts 