Question

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

Answer 1

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

`[1,1.2],[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.