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Numerade Educator

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Problem 20 Easy Difficulty

Express the limit as a definite integral on the given interval.

$ \displaystyle \lim_{n \to \infty} \sum_{i = 1}^n \frac{x_i^*}{(x_i^*)^2 + 4} \, \Delta x $,
[1, 3]

Answer

$$\int_{1}^{3} \frac{x}{x^{2}+4} d x$$

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Video Transcript

when rewriting a limit as an integral. We could always start with the interval that we're interested in, and those were going to be the lower and upper limits to the integral. So the integral from 1 to 3 it's how that's red. And then by the time we get Thio except by stars, we can go ahead and replace all of this each of those exa by stars with exes. So that'll be X divided by X squared plus four and then the Delta X There is the same thing or will change to a D X. And so the idea is, this is adding up an infinite number of rectangles whose heights are determined by this whole expression here and who's with is a small amount of X or Delta X, and we transform that into this notation as an integral