How to Identify and Draw Left, Right and Middle Riemann Sums

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  • 0:10 Review of Riemann Sums
  • 3:58 Mid-Point Riemann Sum
  • 6:25 Left-Sided Riemann Sum
  • 9:28 Right-Sided Riemann Sum
  • 10:48 Lesson Summary
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Lesson Transcript
Instructor: Kelly Sjol
To overestimate or underestimate? In this lesson, you will draw Riemann rectangles so the right side, left side or middle of the rectangle hits the curve.

Review of Riemann Sums

Imagine this. You're sitting there minding your own business, and I throw a sheet of paper in front of you and say, 'What is the area under this curve?' The curve is y=x^2 + 1, and I'm asking about the area between x=0 and x=2. What do you do? Luckily, you remember that the Riemann Sum will give you the area between some function and the x-axis. And you know that the Riemann Sum is nothing more than the sum over k=1 to k=n of f(x sub k) times delta(x sub k). All you're doing is adding up the areas of n slices. Each slice is f(x sub k) tall - that's f(x) tall for the kth slice - with a width of delta(x sub k). So you look at me and say, 'I'm going to use a Riemann Sum. In particular, I'm going to use a Riemann Sum with one slice to make it easy on myself. You just threw a sheet of paper in front of me. You're a little bit crazy.'

The areas of the slices differ based on the location of the selected point
Riemann Sum Different Areas

So you take your Riemann Sum with one slice, and you write this sum as being f(x sub 1) * delta(x sub 1). For f(x sub 1), you pick the left side of this graph and say f(x sub 1)=f(0)=0^2 + 1, which is equal to 1. delta(x sub 1) is the width of your one slice, which is just 2 because it's the right-hand side, 2, minus the left-hand side, 0. You plug in 1 for f(x sub 1) and 2 for delta(x sub 1), and you get 1(2)=2 is the area under this curve. I say, 'No good. Try again!' You say, 'Okay, fine.'

You again take a Riemann Sum with one slice, but this time, you pick your f(x sub 1) to be in the middle of the slice. In the middle of the slice, at x=1, f(x sub 1)=f(1)=1^2 + 1=2. delta(x sub 1) is still going to be 2, but now, the area is going to be 2, which is the height in the middle, times 2, the width. So the area is 4. Okay, great, give me one more. When you took a point on the left side of your segment, to estimate the area of the slice, you got an area that was equal to 2. When you took a point in the middle of the slice, the area was 4. But what happens if you take a point on the right side of this integral? On the right side of the integral, we're going to call f(x sub 1) the point at which x=2, so that's the place where I'm going to measure the height of my first area. So f(x sub 1) is going to equal f(2)=2^2 + 1=5. The width of my slice is still equal to 2. I've got a height of 5 and a width of 2, and I get an area of 10.

When I picked a point on the left-hand side, the area was 2. When I picked a point in the middle, the area was 4, and when I picked a point on the right side, the area was 10. Even though for each one of these I was only doing one slice of the curve. This defines what we know as left-sided Riemann Sums, where you pick the left point of each slice, middle or mid-point Riemann Sums, where you pick the middle of each slice to find an area and right-sided Riemann Sums, where you pick the point on the right side to evaluate the area of a slice.

Finding the mid-point Riemann Sum with two slices
Midpoint Riemann Sum

Mid-Point Riemann Sum

Now you turn around and throw a sheet of paper at me and say, 'What's the area?' Here, your curve is y=x^3 and x is going to go from 0 to 4. I know how this game works, so I ask, 'What kind of Riemann Sum do you want?' and you say, 'I want the middle, or mid-point, Riemann Sum with two slices.' So I divide this up into two segments: one from 0 to 2 and one from 2 to 4. I know that my n in my Riemann Sum is going to be 2, because I have two areas I'm going to add together to get an estimate for the total area. I can expand this Riemann Sum as f(x sub 1) * delta(x sub 1), the area of my first slice, plus f(x sub 2) * delta(x sub 2), the area of the second slice. Because I'm using the mid-point Riemann Sum, f(x sub 1) is going to be taken in the middle of this first slice, at x=1.

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