Question

How do you find the Riemann sum for `f(x) = x - 5 sin 2x` over 0 <x <3 with six terms, taking the sample points to be right endpoints?

Answer 1

`f(x)=x-5sin2x`, `a=0`, `b=3`, `n=6`

Apply the formulas (the ideas).

Cut the interval `[0,3]` into six equal pieces. Each has length
`Delta x = (b-a)/n = (3-0)/6 = 1/2`

Starting at `0` (the leftmost left endpoint), find the right endpoints by successive addition of `Deltax` (the length of the subintervals):

`0+1/2 = 1/2` is the first right endoint

`1/2+1/2 = 1` is the second right endoint

and so on to get:

`1/2, 1, 3/2, 2, 5/2, 3` are the right endpoints.

The Riemann sume is built by finding `f` at the endpoint times `Delta x` and adding:

`f(1/2)*1/2+f(1)*1/2+f(3/2)*1/2+f(2)*1/2+f(5/2)*1/2+f(3)*1/2`

Now do the arithmetic. (You'll either need a calculator or you'll need to accept your answer with sin(1), sin(2), sin(3) etc. in it.)