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Riemann Sums and The Trapezoidal Rule
Riemann Sums, you remember these little guys? The first time you saw them was when you first began doing integrals. A Riemann sum is a way of approximating the area underneath the curve by breaking it up into sections. Sometimes the sections are rectangles, sometimes they are trapezoids.
So you did a bunch of work on Riemann Sums, you struggled, you fought with them. Then you learnt how to do integrals the quick way, and you completely forgot about Riemann Sums. We are here to pat your back up today. So we are got a lovely joke for you here.
What do you get when you divide a jackal hunter by pie? Pumpkin pie.
Upper and lower Riemann Sums Riemann Sums come in a lot of different flavors. Now you might not actually have to calculate any of them in the AP test you never know. But if you do, you need to be prepared. You need to know what the different types are. Because in another kind of question you might be asked are ones that are just asking you to compare the relative sizes of the Sums. First we are going to tackle the upper and lower Riemann Sums.
Remember how a Riemann sum, what you did was you divided the shape up into sections, vertical sections. A lot of times you picked even sections maybe with a width of 1 or a width of 2. They don?t really have to be even though. I can just pick anything I want. I?ll take this point here. I?ll take that point, I'll take that point and I?ll do this one right here. And then draw it vertically upwards the function. If you were actually going to do the calculations, what you?d have to do then is find out how high this is, because you are going to figure area of sub sections.
For the upper Riemann sum what you did then, was you drew your rectangles, so they were all above the function. Going from here to there gives you one. Here to there and then down gives you your next rectangle, over here to there next rectangle. So we?ve divided the interval between here and here into three different rectangles. If you find the area of all those rectangles, and add them up you?ve got a really bad approximation for the area underneath the curve.
That was in the upper Riemann Sums, it?s too big. The lower Riemann sum uses the same spots. But instead of drawing a rectangles for above, we draw them below. We are going to have to use this one this time. Because for the lower Riemann Sum, they are going to be drawn on the underside. I?m going to start over here though to make more sense.
For this one here you would draw underneath and that?s one of the sections we are doing to area on. Then you draw underneath this one, there?s the next section. For the next one, notice I?m using the left side this time. I have to use the left side and my rectangle actually doesn?t even have any height, because this is right at 0,0.
But regardless, if you find that the areas of those two rectangles, and add them up you?ve got an approximation for the area underneath the curve. And this time it?s a really bad approximation. Left and right Riemann Sums are just a different way to name the ones that are made by doing rectangles above or below. But this time instead of worrying whether the section is above or below, you just look to see if you are going to fill in from the left side or the right side.
Notice this time I?ve got a different function. This one is underneath the x axis instead of above.
Let me take a couple of intervals. Draw in the vertical lines. If I was to do that the upper sum, I?d be doing rectangles that all fit above. If I did the lower sum, I do rectangles that fit below. For the left sum, you do your rectangles based on the left side. So based on the left side, for this particular section, its left corner is here. In this section its left corner is here. And in this section its left corner is at 0,0. So it?s a rectangle that again has no height.
That one will be called a left sum if you are suing this definition. It will be called an upper sum if you are doing this. So don?t make the mistake of thinking that the left or right sums are always larger or smaller. You need to actually look where the function is to see which of these is larger or smaller. In this case the left sum is going to turn out to be the one that's smaller than the right sum.
When I do the right sum, you take your rectangles. But this time you are draw them from the right hand side. So you get a bigger rectangle with more area another bigger rectangle with more area. And another bigger rectangle with more area. So in this case, if you are asked to compare them, you would say that the left Riemann sum is smaller than the right Riemann sum. Left Riemann sum is an underestimate, and the right Riemann sum is an overestimate.
The next way you can do a Riemann sum is as a midpoint a midpoint sum. Let?s take a look at one of those. Riemann sum we are still drawing rectangles, we are still going to find their areas. We are still going to add up those areas to get an approximation for the area underneath the curve.
I?ll pick my subsections and this time I will go and find the midpoints. Instead of drawing from the ends of the interval, you need to go to the midpoint. I?m going to draw this as a dotted line so it doesn?t get too confusing because these are going to be the rectangles. Middle between 0 and 2 is 1. Draw upwards. That is going to be the height. If you are actually going to calculate this, you would have to do the average of these in points and then put that result into the formula to find the height of the first rectangle that we are using for the midpoint Riemann Sum.
Next one, between 2 and 4 us 3, draw it straight upwards. If you find that height, it will be the height of the rectangle from the second subsection. And that is the next part of the point Riemann sum. Next between 4 and 8 the average of those two. Remember that?s the quick way to find the midpoint. Just do the average. 4 plus 8 over 2 is 6. Draw a straight up. That location is the height of the next rectangle. Draw across and we have the next section in the midpoint Riemann sum.
In this case, the upper sum would be too high, the lower sum would be too small. The midpoint sum is more like the goldilocks of Riemann Sums. It?s just right not quite just right but it is a lot better.
Notice this area right here is overestimate, but it?s kind of being stopped by this underestimate. Over estimate, underestimate. Overestimate underestimate.
So this one is probably a fairly good approximation for the area underneath the curve. But don?t make this mistake. Don?t assume that the midpoint sum is the average of the upper and lower sums. It usually isn?t.
Trapezoidal sums are the next kind of sums that you can do. Now these aren't Riemann Sums. They have a different way of organizing your graph so that you get usually a pretty good estimate of the area underneath the curve. Trapezoidal sums. It is what it says. You actually do trapezoids and don?t feel like you are stopped, don?t feel like you are trapped. It?s just a trapezoid, you?ve been doing them since geometry.
But you start by picking out your intervals. Same way I did before, pick your intervals. And if you are actually going at doing the trapezoidal sum, you would find the coordinates, the endpoint, the x and y coordinates.
Instead of drawing rectangles we are going to draw trapezoids. Trapezoids also have parallel sides. Here is a set of parallel sides. But the other side?s aren?t necessarily parallel. There is the next side of the trapezoid, and straight between here, and here is the last side of the trapezoid. The area of the trapezoid is ½ sum of the bases times the height of the trapezoid.
The nice thing here is that since we are doing these sums by drawing vertical lines, vertical lines are always perpendicular to the x axis. Like easy a couple of things. It guarantees that these are parallel and it also tells you that width of the interval is really just the height of the trapezoid. Now that can be confusing because you are so used to thinking height is up and down, height is up and down.
No it?s not. In geometry it?s not up and down. Height is the distance between bases. But you know we could go on from here, if we found this height, we would have base 1. If we found this height we have base 2. We could substitute in to the formula, and we would have the area of that sub section. Let?s draw in the other subsections now.
Between here and here it?s not a trapezoid, it?s actually a triangle. But you could actually just use the trapezoid formula. This base right here will be called base 2 and this location isn?t a line. It?s just a point. But you could still call it base 1 and say that the length of base 1 is a 0.
Now here is something that is interesting that?s going to happen. What this would do is show up later on when we make a combined formula for the trapezoidal sum. But just keep this in mind, this section right here, and this section right here, share a side.
Base 2 for this section, is base 1 for this section. That must make a condensed formula later on. Next section looks like this. Next section looks like this, so you have 4 trapezoids, and in this case it?s going to be a really good approximation for the curve. It?s still a slight underestimate. Because there is the area above all the trapezoids, but not too bad. Trapezoidal sums again they aren?t always underestimates. Sometimes they can be overestimates.
Time for a better practice. The AP tests very often will ask you to deal with trapezoidal sums. Recent practice test that have been issued tend to show a lot of trapezoidal problems. We are going to concentrate on those in this section.
So this problem asked you to do use a trapezoidal sum with three sub intervals to estimate area underneath the curve between t equals 0, and t equals 3.
This is something that you might see on free response section. If you do, make sure that you do good notes of what you are dealing with. Because they are going to try to check for your understanding. And to show that you understand this in your graph, you need to show some work.
So I?m going to start off with the graph, and that should go between 0 and 3. Doesn?t have to be a really pretty graph as long as they can read it, that?s all they really care about. 1, 2, 3, put a scale on the graph always want a scale. Where x is and x and y and the function values go between 0 and 1.68. There is something missing there, we'll have to deal with that in a second. Maybe we should deal with that first because we need to know how tall to make our graph. We need that function value.
They may give you all of them, or they may ask you to calculate some. It?s really not that hard. You just put 2.5 into the formula. So you would have 2 sine of 2.5. Calculate your value. I worked that one in advance f(2.5) is about 1.20.
Notice all these decimals. If it?s got these kind of decimals its probably going to be on a calculator section of the test. And you might have a trapezoidal sum on non calculator section, but then they're going to make the numbers easy to deal with.
So I have to go as high as 1.68, only this time I think what I?ll do is, I?ll make each subsection worth 0.5. That way we can make the graph a little bit bigger. A little bit easier to read. You don?t have to have even intervals, even scale from top to bottom.