Midpoint rule - approximating definite integrals, Mathematics

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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.

1951_Midpoint Rule - Integration Techniques.png

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*)]


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