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If your CAS can draw rectangles associated with Riemann sums, use it to draw rectangles associated with Riemann sums that converge to the integrals in Exercises. Use $n=4,10,20,$ and 50 subintervals of equal length in each case.$\int_{1}^{2} \frac{1}{x} d x \quad$ (The integral's value is about 0.693 )

Calculus 1 / AB

Chapter 5

Integrals

Section 3

The Definite Integral

Integration

Campbell University

Harvey Mudd College

Baylor University

University of Michigan - Ann Arbor

Lectures

03:09

In mathematics, precalculu…

31:55

In mathematics, a function…

01:31

If your CAS can draw recta…

04:17

02:47

03:28

03:51

01:42

04:07

01:58

01:45

01:57

okay. What we want to do is we want to show using a graphing utility types that the inner Groll from 1 to 2 of one over x t x um approaches 0.693 And we want to do it by doing the Riemann sums for in equal to 4 10 2050 And so what that means is, as we increase the number of rectangles that we divide the area into that that value the values will approach this 0.69 three. And so we're gonna switch to a graphing utility. I have, um, Dismas, which is a free online tool that you can sign up for, and I'm going to change it toe one divided by experience. My function and we want to go from the left in point was a positive one. The right in point was a positive coup. So we're only interested in this region right here, and the number of intervals is the number of rectangles or partitions. So we want to start with four partitions are now. I have it set up where I am actually doing the right hand. Um, some Cassie's one, which is going to for a decreasing function. The right hand. Some is actually gonna be my lower son. I can actually change this to do the left hand some which is going to give me my upper some. And so let's go ahead and do that. And so I notice for four rectangles or four partitions. My upper some is actually point um 0.759 five and we want to see is if I increase the number rectangles. So my change this to 10 that does my value. My integral approximation. Does it approach 0.693 and so it is getting smaller. So I went from 0.795 2.718 Now let's see if we change it to 20 we get 200.7058 So it is decreasing on a let's do 50 right angles. So sure enough, it went around 2.6 98 And so it is converging as I increase the number of rectangles as I'm increasing the number of rectangles and a mark partitioning my upper. Some is approaching 980.693

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