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Current time:0:00Total duration:7:09

so right over here we have the graph of F and then we have four different expressions and what I encourage you to do is pause this video and see if you can figure out which of these expressions would give the largest quantity the second largest quantity the second smallest quantity and the smallest quantity so I'm assuming you have paused the video and you've given an attempt now let's work through this together so this first expression right over here we're taking the sum from I equals 0 to 9 so we're actually taking the we're actually taking the sum of 10 things because we're taking the 0 thing first second third all the way up to 9 so there's actually going to be the sum of 10 things because we're starting at 0 and we started F of negative 5 plus 0 and so we're going to take so it's going to be we're taking so we're staying negative 5 F of negative 5 plus 0 so that's this height right over here that's that height right over there so we're going to take that and then when I equals one it's going to be negative 5 plus 1 which is negative 4 so it's that height right over there and then negative 5 plus 2 when I equals 2 so it's going to be that height right over there so we're essentially going to sum up all the way so this is going to be negative 5 negative 5 all the way up to so negative 5 all the way negative 5 plus 9 is going to get us all the way to 4 so it's going to be all the way over there all the way over there and so you might be guessing well how do I relate this you know they've already kind of made us think that we're going to somehow relate this to area somehow but how do we actually make that relationship because right now as this is written it's just giving us a buncha essentially a bunch of the values of the functions at different points I guess you could say it's a lot of these heights right over here but one thing that you might that might jump out at you is you could construct rectangles all that have width 1 and so if you multiply the height times the width the area is going to be the same thing as the height so if we put a 1 times 1 right over here this makes it very clear that you're taking the height the width of this rectangle and then this rectangle so you essentially have a bunch of left-handed left-handed rectangles that you could imagine are trying to estimate that's this bluish this bluish area that was shaded in and it's clearly going to be an underestimate because it's giving up it's giving up these areas it's giving up those areas right over there all of these rectangles are sitting there either just touching or they are below the actual function so let me just write this right over here this is going to be an under estimate under estimate of the area of this blue area now let's think about what this one is here so this is f so it's the same thing that we're taking the sum of we're starting at I equals 1 and we're going at 10 so once again 10 things so negative 5 plus 1 is negative 4 F of that is this line right over here is that line right over there and so it looks like it looks like we're taking right-handed rectangles because we could say x 1 so x 1 would be the area if obviously if we multiply by 1 we're not changing we're not changing the value of this expression so that would be the area of this first right-handed rectangle then when I is equal to 2 it's going to be F of negative 3 and so you're looking at this one right over here and so I can let me draw at least this part of it so it's going to look something like this it's going to look something like this where now we are dealing with right-handed rectangles and these are clearly an overestimate these are right-handed rectangles so you're going to go all the way when when I is equal to 10 negative 5 plus 10 is 5 f of 5 that's this line right over here or this length right over here F of 5 and of course we're multiplying it by 1 so it's going to look like that and we could keep going you think you get the general idea now these are all going to be right-handed well these are all right-handed rectangles that I've drawn and these are going to be an overestimate of the area because they all have this little extra they all have this extra region right over here so these are going to be an over over estimate now let's think about this one right over here this is we're going to start at I equals 1 and we're going to go to 20 and it looks like we're going to do rectangles instead of with 1 we're going to do with 1/2 and once again since we're starting at I equals 1 these look like right handed rectangles again this is that we're going to use to estimate so let me do this in a color that you can actually see I'll do this in orange I'll do this in orange right over here so the first one's going to be negative 5 plus 1/2 so it's going to be this F of that which is going to get take you right over there and they're going to multiply it times the width which is 1/2 so now we have twice as many twice as many right-handed rectangles so it's going to look it's going to look like this twice and I won't do all of them because it takes some time we're going to have twice as many right-handed rectangles so it's still going to be an overestimate over estimate but it's going to be less of an over estimate than this one over here because this one over here you had all of this extra green space above the function now we have a lot less extra space a lot less extra space above the function so it's a better estimate but it's still going to be an overestimate because at least for this function right over here because at least over over this interval where the function is increasing the right-handed the right-handed rectangles giving us over estimate but this is a little bit more precise because we're doing we're using we're using narrower rectangles to estimate over estimate but less so but less so then this one right over here and this right over here this is the definite integral from negative 5 to 5 of f of X DX and you can imagine this is essentially the limit as we take these widths to be smaller and smaller and smaller and we essentially end up having essentially an infinite number we approach an infinite number of these right over here and this is the actual area so this is what's actually depicted in blue this is the actual area so if I were to list this from la are just to smallest the biggest over estimate this is the this is the biggest over estimate right over there now this is still going to be an over estimate but it's a little bit more precise because we have more rectangles so I'll put that too now this is the actual area so I'll put that three and then this is actually an underestimate so I would rank this for this is the largest of the values and this is going to be the smallest of the values