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Current time:0:00Total duration:6:45

Riemann approximation introduction

AP.CALC:
LIM‑5 (EU)
,
LIM‑5.A (LO)
,
LIM‑5.A.1 (EK)
,
LIM‑5.A.2 (EK)
,
LIM‑5.A.3 (EK)
,
LIM‑5.A.4 (EK)

Video transcript

what we're going to try to do in this video is approximate the area under the curve y is equal to x squared plus 1 between the intervals X is equal or between the interval x equals 1 and x equals 3 and we're going to approximate it by constructing 4 rectangles under the curve of equal width so let's first think about what those rectangles look like so four rectangles of equal width so look like that like that and like that and I haven't really defined the top of the rectangles just yet but let's think about what those widths have to be if they're going to be equal width and we can call that with Delta X so this distance right over here we're going to call that Delta X so Delta X is going to have to be the total distance that we're traveling in X so we finish at 3 we started at 1 and we want for equal width rectangles so it's going to be equal to 1/2 so for example this first this first interval between the boundary between the first rectangle on the second is going to be 1.5 then we go 1/2 to 2 then we go to 2.5 and then we go 1/2 to 3 now let's think about how we'll define the height of the rectangles for the sake of this video we'll see in future videos that there's other ways of doing this I'm going to use the left boundary of the rectangle to define the height or the function I should say I'm going to use the function evaluated at the left boundary to define the height so for example for the first rectangle this point right over here is f of 1 F of 1 and so I will say that that is the height of our first the height of our first rectangle then we go over here the left boundary of the second rectangle we're now looking at the function evaluated the function evaluated at 1.5 so that is f of 1.5 that's the height and so we get our second rectangle then we get I could keep going like this we get for this third rectangle we have the function evaluated at ooh so we have the function evaluated at to wait so that's right over here that's F of two and so then we get our third rectangle and then finally we have our fourth rectangle the function evaluated at two point five so the function evaluated at two point five is the height so this is f of two point five remember in each of these I'm just looking at the left boundary of the rectangle and evaluating the function there to get the height of the rectangle now that I set it up in this way what is the total approximate area using the sum of these rectangles and clearly this isn't going to be a perfect approximation I'm giving up on a bunch of area here let me see if I can color that in with a color that I have not used so I'm giving up I'm giving up this area I'm giving up this area I'm giving up that area I'm giving up that area there but this is just an approximation and maybe if I had many more rectangles it would be a better it would be a better approximation so let's figure out what the areas of each of the rectangles are so the area of this first rectangle is going to be the height which is F of 1 F of 1 times the base which is Delta X the area of the second rectangle is going to be the height which we already said is f of 1.5 f of 1.5 times the base times Delta X the height of the third rectangle is going to be the function evaluated it's left at let it's left boundary so F of 2 so plus F of 2 times the base times Delta X and then finally the area of the third rectangle is the function the height is the function evaluated at 2.5 so plus that's a different color than what I wanted to use let me I want to use that orange color so plus the function evaluated at 2.5 times the base this is going to be equal to our approximate area let me write make it clear approximate approximate area under the curve just the sum of these rectangles so let's evaluate this so this is going to be equal to f of is going to be equal to the function evaluated at 1 1 squared one is just two so it's going to be 2 times 1/2 plus the function evaluated one point two five one point two five squared is two point two five and then you add one to it it becomes three point two five so plus three point two five times one half times one half and then we have the function evaluated at 2 well 2 squared plus one is five so it's five times one half and then finally you have the function evaluated two point five two point five squared is six point two five times one half plus or six point two five plus one so that's seven point two five seven point two five times one half and just to make the mat simpler we can factor out the 1/2 so this is going to be equal to alright one half in a neutral color 1/2 times 1/2 times two plus three point two five plus three point two five plus five plus five plus seven point two five plus seven point two five which is equal to 1/2 1/2 times let's see if I can do this in my head two plus five is easy that's seven three plus seven is 10 and then we have 0.25 plus 0.25 so it's going to be ten point five plus seven is seventeen point five so 1/2 times seventeen point five which is equal to eight point seven five which once again gives us an approximation clearly the way I've drawn it right over here or though for the function we're using it's going to be an under estimate because we've given up all of that pink area that I had colored in before it's an underestimate but it's an approximation of the area under the curve in the next few videos we're going to try to generalize this to situations where we have an arbitrary function and we have an arbitrary number of rectangles and we'll also start and videos after that we'll look at rectangles where we define the height not by the left boundary but by the right boundary or by the midpoint or maybe we don't use rectangles at all maybe we might things like trapezoids anyway have fun