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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.]

Since A was any real number greater than $1,$ it follows that any Riemann sum can be madearbitrarily 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 $(0,1]$ .

02:04

Frank L.

Calculus 1 / AB

Chapter 5

Integrals

Section 2

The Definite Integral

Integration

Missouri State University

Campbell University

Harvey Mudd College

University of Michigan - Ann Arbor

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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

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