Question

How do you use Riemann sums to evaluate the area under the curve of `1/x` on the closed interval [0,2], with n=4 rectangles using midpoint?

Answer 1

I will use what I think is the usual notation throughout this solution.

Explanation:

`int_0^2 1/xdx`

Note that `f(x) = 1/x` and `a=0` and `b=2`

`n=4` So `Deltax = (b-a)/n = (2-0)/4 =1/2`

All endpoints: star with `a` and add `Deltax` successively:

`0` `underbrace(color(white)"XX")_(+1/2)` `1/2` `underbrace(color(white)"XX")_(+1/2)` `1` `underbrace(color(white)"XX")_(+1/2)` `3/2` `underbrace(color(white)"XX")_(+1/2)` `2`

All endpoints: `0`, `1/2`, `1`, `3/1`, `2`

The subintervals are:

`(0,1/2)`, `(1/2,1)`, `(1, 3/2)`, `(3/2,2)`

We have been aske to use the midpoint of each subinterval as its sample point. The midpoints may be found by averaging the endpoints of each subinterval or by averaging the endpoints of the first subinterval to find its midpoint and then successively adding `Delta x` to get the others.

The midpoints are:

`1/4`, `3/4`, `5/4`, `7/4`

Now the Riemann sum is the sum of the area of the 4 rectangles. We find the area of each rectangle by

`"height" xx "base" = f("sample point") xx Deltax`

Here we are using midpoints for sample points. So

`R = f(1/4)*1/2+f(3/4)*1/2+f(5/4)*1/2+f(7/4)*1/2`

`= (f(1/4)+f(3/4)+f(5/4)+f(7/4))1/2`

Since `f(x)` is the reciprocal of `x`, we have

`=(4/1+4/3+4/5+4/7)*1/2`

Finish the arithmetic to finish.