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Good day students will come two-part one on rectangular approximation so we can also called the Reimann rectangular approximations were going to be looking at how to approximate areas using the left endpoint the right endpoint and of the need points for time approximations alright so before we go ahead and look at the example uhm lets of take a look at the formulas right so the formula is as follows for this down formulas so to approximate to approximate the area under the function F of X on the F of X from X sub one two X up and which is an interval on the X axes okay now the area under the function from X to X X sub one to X of an can be written using the tools of calculus as the definite integral from the lower bound X sub one to the upper bound X up and of the function BX okay so this is the same thing as the area under the function F of X from X sub one to X sub and okay right now how can we approximate this will this can be approximated with and equal sub interval's using the left can rip perfect rectangle approximation L DT is the left and points to the approximation it wanted to the L Graham that is equal to H of times F of X one plus F of X two plus... F of X sub and minus one okay now what we're doing is the recalculating the area of and rectangle okay so eight represents the highest of the eight represents the with of the rectangle and F of X sub one of X up to represents the height okay now since we have uniform let's given by H then we can factor out all the uniform with from all the height okay so used factor them out and then we have this formula right here right one thing to note is that when you're doing your (rectangle approximation use you take all the points on the interval to the left of your right endpoint to the right endpoint is X up and so although uncalibrated points to the left of that starting from X of one we use in computing you (rectangle approximation that's only have X sub in minus one as the last height okay now how about it wanted to computes the right hand alright endpoint rectangle approximation able want to compute this is similar to the left-hand rectangle approximation the only differences we are going to pick all the points to the right of the left endpoint other interval so we using and equals of interval's here so we can have H the perspective with factored out times F of its are starting from the left endpoint we're going to skip that what the weather go straight to F of X up to plus F of X sub three plus the go all the way to the last endpoint okay which is F of X okay to this is your right hand rectangle approximation now lets look at the midpoint rectangle approximation Amram the midpoint rectangle approximation is given by finding the midpoint of all the calibration is on our and sub interval okay so the midpoint of exponent X of to what happened F of X one plus X two over two five plus the

is the midpoint between X two and X three to X up to plus X sub three over to fall that pattern all the way to the midpoint of the last endpoint say calibration so X sub and minus one plus X up and over to okay so these are the formulas so what is H where H is the length of the inner interval X sub and minus X of one that's how long it is an we dividing it into and equals on interval's over and right to that's what H going to with of your rectangles okay so these are the formulas let's go ahead and apply it to an example so for example one where to find the left rectangle approximation the right rectangle approximation and the midpoint rectangle approximation using the formulas the formula as indicated above here then find be exact area using geometry okay remember these are not exact area they could be cannot to be exact with you the approximations in some cases with certain curves somebody approximations my turnouts to be the exact approximation in other cases is not can't so we are going to find the rectangle approximations in then over for the what the exact area is indelibly see how accurate approximations where some of them can one of the minimum turnouts to be one hundred percent accurate okay. So the function under consideration is F of X equals negative X plus four on the interval zero two four using for sub interval's sub interval's okay right so our first dinner going to do here is very going to find the left-hand rectangle approximation so we want to find the area approximate the area under the curve from 0 to 4 this function using the left-hand rectangle approximation sell the area which can be written using calculus notation as the interval the definite integral from zero two four of the

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