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Numerade Educator



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.]


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 $(0,1]$ .

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Video Transcript

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