Question

Suppose f(x)= cos (x). How do you compute the Riemann sum for f(x) on the interval [0, (3pi/2)] obtained by partitioning into 6 equal subintervals and using the right hand end points as sample points?

Answer 1

`R RS = -1.3408`

Explanation:

We have:

`f(x) = cosx`

We want to calculate over the interval `[0,(3pi)/2]` with `6` strips; thus:

`Deltax = ((3pi)/2-0)/6 = (3pi)/12 = pi/4`

Note that we have a fixed interval (strictly speaking a Riemann sum can have a varying sized partition width). The values of the function are tabulated as follows;

Right Riemann Sum

`R RS = sum_(r=1)^6 f(x_i) \ Deltax_i`
`" " = 0.7854 * (0.7071 - 0 - 0.7071 - 1 - 0.7071 - 0)`
`" " = 0.7854 * (-1.7071)`
`" " = -1.3408`

Actual Value

For comparison of accuracy:

`Area = int_0^((3pi)/2) \ cosx \ dx`
`" " = [sinx]_0^((3pi)/2)`
`" " = sin((3pi)/2)-sin(0)`
`" " = -1-0`
`" " = -1`