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Approximating Area Using Rectangles - Problem 2
PhD. in Mathematics
Norm was 4th at the 2004 USA Weightlifting Nationals! He still trains and competes occasionally, despite his busy schedule.
Estimating the area under a curve can be done by adding areas of rectangles. The area of a rectangle is A=hw, where h is height and w is width. To find the width, divide the area being integrated by the number of rectangles n (so, if finding the area under a curve from x=0 to x=6, w = 6-0/n = 6/n.
The height of the rectangle will be f(a) at whatever number a the rectangle is starting. This depends on whether you are taking a left-hand or right-hand sum. For a left hand sum, the height will be taken from the leftmost part of the rectangle. So, if starting at x=0, the height of the first rectangle will be f(0). The height of the next rectangle will be f(0+w), and so on. For a right hand sum, the height will first be taken from the right side of the rectangle. Going back to the example, the height of the first rectangle in a right-hand sum will be f(0+w).
The area under the curve can be approximated by adding the areas of the rectangles. The left-hand and right-hand sums may be different.
Let's do another problem where we estimate the area under a curve using rectangular sums. Let's estimate the area under the curve f(x) equals x² plus 1. From x equals 0 to x equals 2, with left and right hand sums for n equals 4.
N equals 4 first of all means I'm going to use four rectangles. Let me first figure out what the width of the rectangles is going to be. I call the width delta x. The width of the entire interval which is 2 minus 0 divided by the number of rectangles, 4. So that's going to be 1/2, 0.5.
So first I'm going to do our left hand sum. That's actually what you see depicted here. A left hand sum is an estimate that uses rectangles that reach up, and touch the curve in the upper left corner. So each of these rectangles touches the curve in its upper left corner. I can label these values too. If delta x is 0.5, this is 0.5 this is 12, this is 1.5. Those are important numbers.
So the first rectangle is going to have area delta x times f(0). F(0) is the height of this first rectangle. F(0) times delta x plus this one has a height f of 0.5 times delta x. The third one is f(1) times delta x. You can probably see the last one is f(1.5) times delta x.
Now I can factor the delta x out and I have f(0) which with my formula is just 1. F(0.5), 0.5² is 0.25 plus 1 is 1.25. F²(1), 1² is 1 plus 1 is 2. F(1.5), 1.5² is 2.25, plus 1 is 3.25. Then my delta x on the outside is 0.5. So I can't forget that.
Let me add these up on the inside. I have 3 plus 4.5, 7.5 times 0.5. This is going to be 3.75. The left hand sum is 3.75. You can see from the way these rectangles are drawn that this is going to be an underestimate of the area under the curve from 0 to 2.
Let's take a look at our right hand sum. So this picture shows a right hand sum. Here the rectangles are reaching up, and touching the curve on the upper right corner. Same idea, I need to use delta x is 0.5, 2 minus 0 divided by the number of rectangles, 4. The right hand sum is going to be the sum of the areas of each rectangle.
So the first rectangle let me mark these in 0.5, 1, 1.5. The first rectangle has a height of f(0.5), but in the left hand sum, the first height was f(0). so we start with this number rather than this number in the right hand sum. So f(0.5) times delta x plus the second rectangle has a height of f(1), f(1) times 0.5. You can see that this is going to keep going f(1.5) times delta x f(2) times delta x. Now let's calculate these values. I'll pull the delta x out.
F(0.5) again is 1.25 to calculate that before. F(1) is 2. I've calculated that before. F(1.5) is 3.25. We did that one before too. We haven't done f(2) yet. 2² is 4 plus 1 is 5. We still have to multiply by delta x which is 0.5. So let's add these up. I have 7, and 4.5, 11.5 times 5. So what's half of 11.5, 5.75. So 5.75 is my right hand sum.
You can see from the picture that that's an overestimate of the area under the curve between 0 and 2. So If I were to give a good estimate of the area, I'd say that it's somewhere between 3.75, and 5.75. 3.75 is less than the area is less than 5.75. If I wanted to get a better approximation, I could use more rectangles.
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