Problem

Express the limit as a definite integral. $ \d…

01:54

Problem 72 Hard Difficulty

Let $ f(0) = 0 $ and $ f(x) = 1/x $ if $ 0 < x \le 1 $. Show that $ f $ is not integrable on $ [0, 1] $. [$ Hint: $ Show that the first term in the Riemann sum, $ f(x_i^*) \Delta x $, can be made arbitrarily large.]

Name: let f0 0 and fx 1x if 0 x le 1 show that f is not integrable on 0 1 hint show that the first term in
Uploaded: 2019-12-01
Duration: 56 s
Channel: Numerade
Description: Let $ f(0) = 0 $ and $ f(x) = 1/x $ if $ 0 < x \le 1 $. Show that $ f $ is not integrable on $ [0, 1] $. [$ Hint: $ Show that the first term in the Riemann sum, $ f(x_i^*) \Delta x $, can be made arbitrarily large.]

Answer

Since A was any real number greater than $1,$ it follows that any Riemann sum can be made
arbitrarily large. Therefore, the Riemann sums to not have a limit as the largest $\Delta x$ tends to zero. It follows that $f$ is not integrable on $(0,1]$ .

Catherine R.

Missouri State University

Anna Marie V.

Campbell University

Kayleah T.

Harvey Mudd College

Kristen K.

University of Michigan - Ann Arbor

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Video Transcript

We know that if and it's the number of rectangles in the width of the wrecked tank would be Delta Acts is one minus here, over on which is one over an There for the area of the first rectangle would simply be, and therefore we know we have the sum from aye equals one. Don't forget the limit. We know this is greater than the limit. As UN approaches, infinity on equals infinity as you can see the first term and some go to infinity. So it's not Integra ble on a certain interval if it goes towards positive or negative infinity. Therefore there's no limit, so it's not in terrible on this interval 01

Amrita B.

Related Courses

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Calculus: Early Transcendentals

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Integrals

Section 2

The Definite Integral