Question

How do you use Riemann sums to evaluate the area under the curve of `f(x)= In(x)` on the closed interval [3,18], with n=3 rectangles using right, left, and midpoints?

Answer 1

Please see the explanation section below.

Explanation:

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

Note that `f(x) = lnx` and `a=3` and `b=18`

`n=3` So `Deltax = (b-a)/n = (18-3)/3 =5`

The endpoints: start with `a` and add `Deltax` successively:

`3` `underbrace(color(white)"XX")_(+5)` `8` `underbrace(color(white)"XX")_(+5)` `13` `underbrace(color(white)"XX")_(+5)` `18`

The subintervals then are:

`[3,8]`, `[8,13]`, and `[13,18]`

The left endpoints are: `3`, `8` and `13`.

The right endpoints are `8`, `13`, and `18`

The midpoints may be found by averaging the endpoints.
They are: `5.5`, `10.5`, `15.5`
(As an alternative method, find the first midpoint and add `Deltax` successively.)

Now the Riemann sum is the sum of the area of 3 rectangles. We find the area of each rectangle by
`"height" xx "base" = f("sample point") xx Deltax`

So, using left endpoints, we have

`L = f(3)*5+f(8)*5+f(13)*5`

`= (f(3)+f(8)+f(13))*5`

The arithmetic is left to the student.

Using right endpoints we have

`R = f(8)*5+f(13)*5+f(18)*5`

`= (f(8)+f(13)+f(18))*5`

The arithmetic is left to the student.

Using midpoints we have

`M = f(5.5)*5+f(10.5)*5+f(15.5)*5`

`= (f(5.5)+f(10.5)+f(15.5))*5`

The arithmetic is left to the student.