Question

How do you find an approximation to the integral `int(x^2-x)dx` from 0 to 2 using a Riemann sum with 4 subintervals, using right endpoints as sample points?

Answer 1

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

Explanation:

`int_0^2 (x^2-x)dx`

Note that `f(x) = x^2-x` and `a=0` and `b=2`

`n=4`

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

All endpoints: start 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`

Right endpoints: `1/2`, `1`, `3/1`, `2`

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("endpoint") xx Deltax`

So

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

`= (f(1/2)+f(1)+f(3/2)+f(2))1/2`

`R = ((-1/4)+(0)+(3/4)+(2))1/2 = 5/2*1/2 = 5/4`