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Riemann sums in summation notation: challenge problem

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- [Voiceover] The graph of F is shown below. A total of 24 right hand rectangles are shown. So, what do I mean by right hand rectangles? So, there's clearly 24 rectangles. You can count them. And right hand rectangle means that for each of these rectangles the height of the rectangle is defined by the value of the function on the right hand side of the rectangle. So you can see this is the right hand side of this first rectangle and if you take the value of the function of that point that is the height of the rectangle. A left hand rectangle would define the height of the rectangle by the value of the function on the left hand side of the rectangle. So, a left handed rectangle's height, the first rectangle's height would look like that. That's what they mean by right handed rectangle Fair enough, eight in blue. We see that. 16 in red. All right. All 24 of the rectangles have the same width. Which of the statements below is or are true? They give us three expressions in sigma notation and they say, like this first one is the sum of the areas of the blue rectangles. This is the sum of the areas of the red rectangles. This is the sum of the areas of all of the rectangles. So, I encourage you now to pause the video and try to determine on your own, which of the statements is or are true. So, I assume you've had a go at it. Let's just go through each of these and see whether they make sense. So this first one, the sum of the areas of the blue rectangles. Well, we know we have one, two, three, four, five, six, seven, eight blue rectangles, and we're summing from one to eight. So it seems like we're summing eight things right over here. This is one, two, three, four, five, six, seven, eight. So, this is looking good right over here. And then we take F of something times one half. So, we're not even looking at this yet. It looks like this would be the height of each of the rectangles. Remember, we're taking the value of the function on the right hand side for the height, and this would be the width. So does it make sense that the width of each of these rectangles is one half? Well, the total distance between X equals negative five and X equals seven, that distance is 12. Five plus seven, that's 12, and we're dividing it into 24 rectangles of equal width. So, if you divide 12 divided by 24 each of these are going to have a width of one half. So this is checking out that the one half. Now let's think about this part. Let's think about the F of negative five plus I over two. So, let's see. When I is equal to one, so we're going to take one half times F of negative five plus one over two. Right? I is one. So negative five plus one half is going to get us to this point right over here. F of that is going to be this distance, this height right over here. This is consistent with these being right handed rectangles. So, this is definitely the case. When I is equal to one we're definitely finding this area right over here. When I is equal to two it's going to be negative five plus two over two. So two over two, we're going to add one, we're going to go over here. So once again we're doing one half, which is this right over here. That's the width times F of negative five plus two over two, which is F of negative four, which is this height right over here. So, once again that is that area. You can keep following it. Everytime we're taking the function this first one is negative five plus one half, and then for each increment we're adding a half to I guess the right hand side is one way to think about it. So, this actually makes complete sense. We're doing this for the first eight so this is indeed true. This is the sum of the areas of the blue rectangles. Now let's look at this one over here. The sum of the areas of the red rectangles. At first this looks pretty interesting. We're taking the sum of 16 things, and we do indeed have 16 things right over here. We have the width of each of those 16 things or for each of those things we want to figure out an area, and it is indeed the case that each of these has a width of one half. But what happens when we take F of negative one plus I over two. So, we're starting right over here at negative one. Negative one plus I over two. When I equals one we're going to be at this point right over here, and F of that is going to be -- You might say, "Hey, isn't that going to be the height of that rectangle? When I equals two isn't it going to be the height of that rectangle? And when I is equal to three isn't it going to be the height of that rectangle?" And that's where we have to be very careful. They're going to have the same absolute value, but these are going to be these are all going to be negative values. These are all going to be negative because we see between this value of our function. So that looks like between negative one half all the way to seven our function is actually negative. So one way to think about it, you'd be having negative height so when you multiply these two things you're going to get a negative number. So this whole thing is going to be a negative number and so you're essentially going to get the negative of the sums of the areas of the red rectangles, but that's not the same thing as the sum of the areas of the red rectangles. An area is, at least in the traditional sense, you would expect to have, if you were just seeing what's the area how much carpet you would need to cover this someone would say that will be a positive value. But this is going to be a negative version of that. So that is not the sum of the areas of the areas of the red rectangles. It's the negative of the areas of the red rectangles. So we'd rule that one out. Then this last choice. So, this expression is the sum of the areas of all the rectangles. And so this one is going from I equals one to 24. So it's literally 24 things. It's starting here and it just keeps going. And if this said from I equals one to I equals eight it would be the first choice, but then this falls into the problem again of once we get past, when we get to I equal nine this thing right over here will turn negative, and it's going to give the negative area. So it's essentially going to net out this positive area against this negative area right over here. So it's not the sum of the areas of all the rectangles. It's going to be this area essentially minus this area right over there.