How to Find the Limits of Riemann Sums

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  • 0:06 Determining the Size of Land
  • 2:22 Using the Riemann Sum
  • 6:21 Defining the Integral
  • 7:38 Lesson Summary
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Lesson Transcript
Instructor: Zach Pino
What would happen if you could draw an infinite number of infinitesimally thin rectangles? You'd get the exact area under a curve! Define the Holy Grail of calculus, the integral, in this lesson.

Determining the Size of Land

Slicing the property into several stripes provides a better estimate of the total area
Riemann sums determining the size of land

Let's go back and say you're trying to calculate the amount of land that you have - the area of your property. Your property extends from some road to the River Newton, between a fire hydrant on the west side and a pine tree on the east side. You know that you can estimate the area by measuring how far it is from the road to the river at the fire hydrant and multiplying that number times the distance between the fire hydrant and the pine tree. That gives you a rough estimate of the area, as it's the height of your property times the width of your property.

You know that you could make a better estimate of the amount of property that you have by dividing this up and calculating two separate areas - say the area from the fire hydrant to the middle of your property line and then measuring again how far back the river is and the middle of your property and multiplying that height times the width from that middle point to your pine tree. Once you have these two separate areas, you can add them together and get a better estimate of your total area.

If you divided your property into three regions and measured the area of each one of them, you might get still a better estimate of your overall area. And really, you could do this over 16 regions. Maybe each one is the width of a lawn mower. So as you're riding on your lawn mower, you measure how many feet it is from the road to the river; you turn around and measure how far it is from the river to the road. And you can continue doing this in big stripes.

This is just slicing your property and measuring the area of each stripe separately. This would give you an even better estimate of your area. When you mow the lawn, you get a pretty good idea of how much land you have. Sure, there's the odd patch of grass that's left over, and maybe every now and then you drive over the road a little bit, but it's a pretty good estimate as to how much land you actually have.

Using Riemann sum notation, the x-axis is the road and f(x) is the river
Riemann Sum Notation and Graph

Using the Riemann Sum

All this is is a Riemann sum. It's the area between the river and the road, and you've added up the sums of all of these little slices. In Riemann sum notation, we call the river f(x), the road the x-axis, the fire hydrant is at x = some value a, and the pine tree is at the x value of b. We write this whole Riemann sum as the sum over all n slices - so that's k=1 to n (from the first slice to the nth) - the height of that slice, which is f(x sub k), the height of slice k, times the width of that slice, that's delta x sub k. So the height times the width, sum them all up and you get a pretty good estimate for your area.

Let's say we have a really funky looking property. Say the river follows the line x^3 - sin(x) + 1, and our property goes from x=0 to x=2. If we take one slice and we estimate this with one Riemann sum, a left-side Riemann sum, we might get an area that is 2. If we use two slices, dividing this in the middle, then we might get an area of 2.16 or so. If we divide this into five slices, our area becomes 3.3. If we divide it into ten slices, it's roughly 3.9, 20 slices is 4.4 or so. If I divide this into 100 slices - say I'm able to take my lawnmower and go up and back 100 times - I might estimate the area to be 4.5.

The smaller the slices, the closer you are to the actual area
Using the Riemann Sum Graph

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