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Approximating Area Using Rectangles - Concept
PhD. in Mathematics
Norm was 4th at the 2004 USA Weightlifting Nationals! He still trains and competes occasionally, despite his busy schedule.
When finding the area under a curve for a region, it is often easiest to approximate area using a summation series. This approximation is a summation of areas of rectangles. The rectangles can be either left-handed or right-handed and, depending on the concavity, will either overestimate or underestimate the true area.
I want to talk about a really important concept in Calculus. The idea of finding the area under a curve. Let me define what that means.
Consider this picture here. I've got a function graph y=f of x and I've defined an interval from a to b. I want to find this area that's shaded in here. So the area under the curve problem is stated as follows. Let f of x be a non-negative function on the interval ab. Find the area of the region lying beneath the curve y=f of x. So below the curve like this and above the x axis. So above this, from x=a to x=b. So between these guys. So whenever I say find the area under the curve y=f of x on the interval ab this is what I'm going to mean. And I'll be dealing with only non-negative functions. No functions that dip below the x axis for now. Finding the area under the curve will imply that we're dealing with a non-negative function.
Now, there are a couple of ways to approach this for starters. And the way we're going to do it is we're going to divide the interval up into sub intervals and construct little rectangles. This is how, this is an important calculus concept. The idea of a rectangular sum, sometimes called the Riemann sum. So let me start by explaining. First, the interval that I'm interested in from a to b, I'm going to divide it into n equal sub-intervals. The actual number of sub-intervals isn't important right now. In future exercises we use like 3, 4, 10, 20. n equal sub-intervals of which delta x. So all these sub-intervals have to be equal in width and I'll call that width delta x.
Now since I can calculate the length of the whole interval from a to b, that's just b-a. I divide by the number of intervals n, and I get the width of each interval, delta x. So wherever you see a delta x, just remember that it actually equals b-a over n. That's the width of every one of these rectangles.
Now, I'm approximating the area under this curve and the way I'm going to do it is using the areas of each rectangle, right? And in this picture I have 4 rectangles. But let's just pretend that it's n rectangles.
So for my first rectangle, my area's going to be width times height. And the height is determined by where the rectangle touches my curve y=f of x. And it does so at the first point at a which I'm going to call x sub 0. So f of x sub 0 is this height. f of x sub 0 times delta x is the area of this rectangle, that's this term in my sum. And the next term is also the area of a rectangle. It's delta x times f of x sub 1. Here's x sub 1. f of x sub 1 is the height of this rectangle. So that height times this width gives me another area. And I'm going to keep doing this for all n of my rectangles until I get to the last one. Notice, in a left hand sum, I'll explain what that term means in a second. Left hand sum, the last term I use is x of n-1. Not this term but the one right before it. That's how I get the height of my last rectangle. So this height here is f of x sub n-1 times the width delta x, that'll give me the area of my last rectangle.
Now sometimes you'll see this sum written this way. The sum, this is the Greek capital letter S, sigma for sum form i=0 to n-1 of f of x of i delta x. What this means is, it's basically this means exactly the same as this.
Each term of this if this sum has the form f of x sub i times delta x just like this. And what I'm doing with this notation here is I'm saying add up all the terms of this kind from i=0 to n-1 incrementing up by 1. So that's what this notation needs, a compact form of this extended sum.
Now why left hand sum? Each of these rectangles rises up and touches my curve on the upper left corner. So it's called a left hand sum because of that. And that's why we use these left hand points x 0, for the first sub-interval, x 1 for the second sub-interval. We'd use x 2 for the third and for the last one we use x f of x sub n-1, alright? That's a left hand sum and of course it stands a reason there's also a right hand sum. And a right hand sum is very very similar.
Same idea in the beginning. We divide the interval from a to b into n sub-intervals and each has the same width, still delta x. We're still going to add up the areas of rectangles only this time, we'll let the rectangle rise up until it touches the curve on the right hand corner. That's what we call a right hand sum.
So the first area is going to be delta x times this height which is f of x sub 1, that's this term here. The second rectangle is delta x times, excuse me f of x sub 2, this term. And we go all the way to the end here and this is our last rectangle whose area is delta x times this height which is f of this this point b which is called x sub n in my partition here. So, these are going to be the n areas of rectangles and this is the sigma notation for the same sum, right? The compact form of this sum. This is a right hand sum but a lot of times I'm going to write it in the expanded form like this so you don't have to worry about the sigma notation. Your teacher may care about this. Anyway, these are both kinds of rectangular sums of Reimann sums that are used to approximate the area under a curve and this is a very important concept in Calculus.
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