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Current time:0:00Total duration:6:36

Riemann sums in summation notation: challenge problem

Video transcript

the graph of f is shown below a total of 24 right-hand rectangles are shown so what do you mean by right-hand rectangle 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 at 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 at the left-hand side of the rectangle so a left handed rectangles height would be the first rectangle would look like that so that's what they mean by right-handed rectangle fair enough 8 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 and they give us three expressions in Sigma notation and they say like this first one is the sum of the areas of the blue rectangle this is the sum of the areas of the red rectangles this is the sum of the areas of all 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've assumed I assume you've had a go at it so 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 1 2 3 4 5 6 7 8 blue rectangles and we're summing from 1 to 8 so it seems like we're summing 8 things right over here this is 1 2 3 4 5 6 7 8 so this is looking good right over here and then we take F of something times 1/2 and so now 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 1/2 well the total distance between x equals negative 5 and x equals 7 that distance is 12 5 plus 7 that's 12 we're dividing it into 24 rectangles of equal width so if you divide 20 12 divided by 24 each of these are going to have a width a a width of 1/2 so this this is checking out that the 1/2 and now let's think about this part let's think about the F of negative 5 plus I over 2 so let's see when when I is equal to 1 so we're gonna take 1/2 times F of negative 5 plus 1 over 2 right I is 1 so negative 5 plus 1/2 is going to get us to this point right over here F of that is going to is going to be this distance this height right over here and with this is consistent with these being right handed rectangles so this is definitely the case when I is equal to 1 we're definitely taking we're definitely finding this area right over here when I is equal to 2 it's going to be negative 5 plus 2 over 2 so 2 over 2 is going we're going to add 1 we're going to go over here and so once again we're doing 1/2 which is this right over here that's the width times F of F of negative 5 plus 2 over 2 which is F of negative 4 which is this height right over here so once again that is that area and you can keep following it every time we're taking the function we're starting it we're doing which this first one is negative 5 plus 1/2 and then for each increment we're adding 1/2 to I guess the right hand side is one way to think about it so this actually makes all makes complete sense and we're doing this for the first 8 so this is indeed 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 what 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 1/2 but what happens when we take F of negative 1 plus I over - so we're starting we're starting right over here at negative one negative one plus five or 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 going to be the height of that rectangle and when I is equal to three is it going to be the height of that rectangle and that's where we have to be a bit we have to be very careful it's going they're going to have the same they're going to 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 between because we see between F 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 what you'd be having negative Heights so when you multiply these two things you're going to get a negative number so these are going to be this 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 all the you know if you were just saying what's the area how much carpet you would need to cover this some will tell you that would 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 red rectangles it's the negative of the areas of the red rectangle so we'd rule that one out and then this last choice so this expression is the sum of the areas of all the rectangles and so this one's two going from I equals 1 to 24 so it's literally 24 things it's starting it it's starting here and it just keeps going and if this said from I equals 1 to I equals 8 it would be the first choice but then this falls into the problem again if once we get past when we gets to I equal 9 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 over there