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How to Use Riemann Sums for Functions and Graphs

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  • 0:06 Determining the Size…
  • 1:26 Using Multiple Areas
  • 3:20 Using Sum Notation
  • 6:15 Understanding Riemann Sums
  • 6:45 Lesson Summary
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Lesson Transcript
Instructor: Robert Egan
Find out how Riemann sums can be used to calculate multiple areas efficiently. In this lesson, you'll learn how this can come in handy for irregular areas and how you can put it to use.

Determining the Size of Your Land

The total area of the property is equal to the sum of the areas of the 16 slices
Property Area Example

You've just inherited a piece of land, and you want to know exactly how much land you have. So how would you determine how much land you have? Let's draw out your plot. Your land goes between a road and a river. The river is kind of curvy; it changes, so it's not always the same distance away from the road. You know that your land extends along the road from a fire hydrant to a pine tree. Everything in there is yours, so you can draw a line from the hydrant to the tree and a line from the pine tree to the river. How would you estimate how much area this actually is? One way you might do it is look at how far along the road your property extends and how far back the river is, say, at the fire hydrant. You've got a width along the road, and you've got a height that's at your fire hydrant. If you look at your property on a map, this may not be the best estimate of your area. You could be missing a region and might even be including some of the river or even the land across the river.

Using Multiple Areas

So how could you get a better estimate? Let's say you measure how far back the river is at the fire hydrant and then you take another measurement, say at the exact middle of your property. That's how far back the river is at the middle of the property. So you find the area of those two regions, with the first region being the area between the fire hydrant and the middle of your property and the second region being the area between the middle of your property and your tree. You also know your total area is going to be the sum of your first area plus your second area. Each of your areas is the height at that point times the width of your property divided by two, because each area is half the width of your total property.

How to use sum notation
Sum Notation Components

Okay, so maybe that's a better estimate of your property. But there may still be an area or two that you're missing, and you're not convinced. Let's say you want a better estimate, so you divide your property into thirds. You do the same thing and say the total area is equal to the first third (near the fire hydrant), the middle third and the last third (by that pine tree). While you're at it, why not divide your property into 16 parts and measure the distance from the river to the road somewhere for each of these 16 parts? Now you have the height and width for each of the 16 parts. Since you have the height and width, you can add up the area of each of the 16 parts to calculate the total area of your property. Well, this is getting to be a lot - I mean, 16 terms, 16 areas that we've got to add up.

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