Midpoint Rule - Approximating Definite Integrals
This is the rule which should be somewhat well-known to you. We will divide the interval [a,b] into n subintervals of equal width that is,
Δx = (b-a) / n
We will indicate each of the intervals like,
[x0,x1],[ x1,x2],......, [xn-1,xn] where x0 = a and xn = b
After that for each interval let xi* i x be the midpoint of the interval. After that we sketch in rectangles for each subinterval with a height of f (xi*). At this point there is a graph showing the set up using n=6.
We can simply find out the area for each of these rectangles and thus for a general n we get that,
∫ba f(x) dx ≈ Δx f (x1*) + Δx f (x2*) + ..... + Δx f (x*n)
Or, on factoring out a Δx we acquire the general Midpoint Rule.
∫ba f(x) dx ≈ Δx [f (x1*) + f (x2*) + ..... + f (xn*)]