Question

How do you find an approximation for the definite integrals `int 1/x` by calculating the Riemann sum with 4 subdivisions using the right endpoints from 1 to 4?

Answer 1

Assuming equal subdivisions, see the explanation section below.

Explanation:

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

`int_1^4 1/x dx`

Note that `f(x) = 1/x` and `a=1` and `b=4`

`n=4` So `Deltax = (b-a)/n = (4-1)/4 =3/4`

To find all of the endpoints: start with `a` and add `Deltax` successively:

`1` `underbrace(color(white)"XX")_(+3/4)` `7/4` `underbrace(color(white)"XX")_(+3/4)` `10/4` `underbrace(color(white)"XX")_(+3/4)` `13/4` `underbrace(color(white)"XX")_(+3/4)` `16/4`

Right endpoints: `7/4`, `5/2`, `13/4`, `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("endpoint") xx Deltax`

So

`R = f(7/4)*1+f(5/2)*1+f(13/4)*1+f(4)*1`

`= 4/7+2/5+4/13+1/4`

To finish, do the arithmetic.