Question

Explain where approximation schemes come from i.e. Midpoint ,Trapezoidal Simpsons

Answer 1

So, im not the best at this explanation but ill try here ...do ya'll remember the pain in the retricecalled Reimann's sums - yeah you should it takes literally all your time to solve , thats why we use Calculus.

So if you've got a nice curve say y=-(x-2)^2 for the sake of simplicity, if your not familiar this is a shift to the right 2 units and reflected about the x axis.)
So with a Reimann's sum the ugly looking formula with summation and delta x and x sub i etc we want to find the Area below this curve , well thats a pain to do right since its such a wide area well why not make it simple and break it u p right?

lets use rectangles ( cus thats what the sum tells us) then our delta x is the width of each , now what? well we cut it up into how many we want once each are the same width apart that gives us the b-a / n business then for midpoint rule its going to be 1/2 (xi-1 +x sub i ) and thus is the midpoint between two points on rectangles.

CAN ANYONE TELL ME FOR SIMPSONS RULE AND THE OTHER ONE ?

Answer 2

Take a look at these: http://is.gd/RvWjqS & http://is.gd/zhH4h1