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Math 104: Calculus16 chapters | 135 lessons | 11 flashcard sets

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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.

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.

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.

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.

Maybe we should use sum notation. Remember that sum notation adds a bunch of terms and labels these terms as 1 to *n*. We write this as the sum from *k* - which is our index of summation - equals 1 (our lower limit) to *n* (our upper limit) of *a* sub *k*. So this is for all of these individual terms. In the case of our property, where we're taking 16 parts, we're going to add up all of those areas. So we're going to add up the area of the first part, the second part, the third part ... all the way up to that 16th part. To find out the area of the property, we're going to write the sum from *k*=1 to 16 of the area of the region *k*. That's the area of that first slice plus the area of that second slice plus the area of the third slice and so on, but written a little more compactly.

Okay, so how would we do this mathematically? Let's call the river *f(x)*. That's how far it is from the road, which we're going to call the *x*-axis. Let's put in a *y*-axis, call the fire hydrant *a*; that's the left-hand side of our property. And we'll call the pine tree *b*; that's the right-hand side of our property. Let's look at one particular area. Say we divide the property into 16 slices, and let's look at the eighth slice. We want to find out what the area is of this eighth slice. The width of this eighth area is going to be some *delta x*. The height of the area is going to be *f(x)* somewhere in this eighth region, so I'm going to call that *f*(*x* sub 8). The area of that eighth slice is going to *f*(*x* sub 8) * (*delta x* sub 8) - I'm calling that width *delta x* sub 8 just so each slice can be a different width if I want it to be. If I were to use this kind of notation for the first area, I would have *f*(*x* sub 1) * (*delta x* sub 1), the second area would be *f*(*x* sub 2) * (*delta x* sub 2), and so on for all 16 areas. The 16th area would be *f*(*x* sub 16) * (*delta x* sub 16). If I wanted to find the total area, it's going to be the sum of all these individual areas. I could write this as the sum from *k*=1 to 16 of the area of slice *k*. If I plug in each of these areas, then I know that the area of *k* is *f*(*x* sub *k*) * (*delta x* sub *k*). So I can write this as the sum of *k* from 1 to 16 of *f*(*x* sub *k*) * (*delta x* sub *k*).

This is what's known as a **Riemann sum**, and it isn't limited to 16 slices. You can divide your area up into *n* slices. Generally, we write Riemann sums as the area between some function, *y*=*f(x)*, and the *x*-axis. Using the sum notation, this is equal to the sum, as *k* goes from 1 to *n*, of *f*(*x* sub *k*) * (*delta x* sub *k*) for each of the *n* slices.

Just think of the **Riemann sum** as a way to estimate the area of your property. All it is is a way to add up all the little different slices of your property from the road to the river. Each of these slices has an area *f*(*x* sub *k*) * (*delta x* sub *k*). When you add them all up, you get the sum from *k*=1 to *n* for all *n* slices of *f*(*x* sub *k*) * (*delta x* sub *k*). This is the Riemann sum.

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Math 104: Calculus16 chapters | 135 lessons | 11 flashcard sets

- Go to Continuity

- Go to Series

- Go to Limits

- Summation Notation and Mathematical Series 6:01
- How to Use Riemann Sums for Functions and Graphs 7:25
- What is the Trapezoid Rule? 10:19
- How to Find the Limits of Riemann Sums 8:04
- Definite Integrals: Definition 6:49
- How to Use Riemann Sums to Calculate Integrals 7:21
- Linear Properties of Definite Integrals 7:38
- Average Value Theorem 5:17
- The Fundamental Theorem of Calculus 7:52
- Indefinite Integrals as Anti Derivatives 9:57
- How to Find the Arc Length of a Function 7:11
- Go to Area Under the Curve and Integrals

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