I. Express each limit as an integral. 1) limn →∞∑k=1^nln(1 +...

Transcript

00:01 In this problem, our job is to use riemann sums with right end points to write the following integral as a limit, the integral from 2 to 5 of x squared plus 1 over x with respect to x.

00:13 So we start by identifying the function to be integrated or the integrand, f of x will be x squared plus 1 over x.

00:24 We know that a equals 2 and b equals 5, the lower and upper limits of integration.

00:29 So that tells us that a typical width, we will use right end points and we will assume that the widths of the rectangles are all the same.

00:38 So the width delta x is simply the length of that interval divided by n, where n is the number of rectangles.

00:46 So our width in each case is going to be 3 over n.

00:51 Now to use right end points, that means our x sub i's, the x's that partition the interval from 2 to 5, we'll start at 2 and for each i, we will add another 3 over n, that is another width.

01:07 So another way to write the x sub i is 2 plus 3i over n.

01:15 Alright, so we will use the delta x and the x sub i to write down the next step.

01:23 So what does it mean to use right end points with these riemann sums? the right end points, the heights of the rectangles, the widths are always delta x, the heights will vary and the heights will be found by taking the right end point of each little sub interval.

01:40 That means f of x sub i.

01:42 So we're going to take our expression for x sub i here and substitute it into the function.

01:48 So we will get the 2 plus 3i over n, we will square it, plus, and we will do 1 divided by the 2 plus 3i over n.

02:00 So we now have the heights and the widths of our rectangle.

02:03 So we will let r sub n denote typical riemann sum using right end points.

02:09 So we want to add up the areas of these rectangles...

Question

I. Express each limit as an integral. 1) $\lim_{n \to \infty} \sum_{k=1}^{n} \ln(1 + \frac{2k}{n}) \cdot \frac{2}{n}$ 2) $\lim_{n \to \infty} \sum_{k=1}^{n} \frac{k^4}{n^5}$

Question

I. Express each limit as an integral. 1) $\lim_{n \to \infty} \sum_{k=1}^{n} \ln(1 + \frac{2k}{n}) \cdot \frac{2}{n}$ 2) $\lim_{n \to \infty} \sum_{k=1}^{n} \frac{k^4}{n^5}$

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