Question

How do you use Riemann sums to evaluate the area under the curve of `f(x) = x^2 + 3x` on the closed interval [0,8], with n=4 rectangles using midpoint?

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) = x^2+3x` and `a=0` and `b=8`

`n=4` So `Deltax = (b-a)/n = (8-0)/4 =2`

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

`0` `underbrace(color(white)"XX")_(+2)` `2` `underbrace(color(white)"XX")_(+2)` `4` `underbrace(color(white)"XX")_(+2)` `6` `underbrace(color(white)"XX")_(+2)` `8`

The midpoints may be found by averaging the endpoints.
They are: `1`, `3`, `5`, `7`

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("midpoint") xx Deltax`

So

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

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

The arithmetic is left to the student.