How do you use the Midpoint Rule with ##n=5## to approximate the integral ##int_1^(2)1/xdx## ?

The interval ##[1,2]## is divided into 5 equal subintervals

##[1,1.2],[1.2,1.4],[1.2,1.4],[1.4,1.6],[1.6,1.8], and [1.8,2]##.

Each interval are of length ##Delta x={b-a}/n={2-1}/5=0.2##.

The midpoints of the above subintervals are

##1.1,1.3,1.5,1.7, and 1.9##.

Using the above midpoints to determine the heights of the approximating rectangles, we have

##M_5=[f(1.1)+f(1.3)+f(1.5)+f(1.7)+f(1.9)]Delta x##

##=(1/1.1+1/1.3+1/1.5+1/1.7+1/1.9)cdot 0.2 approx 0.692##

By Midpoint Rule,

##int_1^2 1/x dx approx 0.692##.

I hope that this was helpful.

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