Question

How do you find the Riemann sum for `f(x)=sinx` over the interval `[0,2pi]` using four rectangles of equal width?

Answer 1

`[sin(x_1 "*") + sin(x_2 "*") + sin(x_3 "*") + sin(x_4 "*")] (pi/2)` where the `x_i"*"` are the sample points.

Explanation:

`[a,b] = [0,2pi]` and `n = 4`

so `Delta x = (b-a)/n = (2pi)/4 = pi/2`

The Riemann sum with `n=4` is:

`[f(x_1 "*") + f(x_2 "*") + f(x_3 "*") + f(x_4 "*")] Delta x`

where the `x_i"*"` are the sample points.

So, for this question we get:

`[sin(x_1 "*") + sin(x_2 "*") + sin(x_3 "*") + sin(x_4 "*")] (pi/2)`
where the `x_i"*"` are the sample points.

The intervals are:
`[0.pi/2]`, `" "[pi/2, pi]`, `" "[pi, (3pi)/2]`, and `" "[(3pi)/2, 2pi]`

so `x_1"*"` is in `[0.pi/2]`

and `x_2"*"` is in `" "[pi/2, pi]`

and so on.

To go any further and get the same answers, we would need to agree on what to use for sample points.