Chapter 5.2, Problem 71E

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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

Explanation

**Given:**

The value of the function *x* is irrational number.

The value of the function *x* is rational number.

**Calculation:**

Consider the function *f* is integrable within the limit

Show the Theorem 4 as shown below:

If *f* is integrable on

Refer to theorem 4.

The value of *f* is integrable on

Consider a positive integer as *n*.

Divide the interval *n* equal interval.

Consider the

The expression to find the Riemann sum

Find the width

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).

Calculate the Riemann sum using the Equation (1).

Substitute

The value of

Calculate the value of

Substitute 0 for

Thus, the value of the

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