Answer
When $~~n = 5~~$, then $~~R_5 = 1.933766$
When $~~n = 10~~$, then $~~R_{10} = 1.983524$
When $~~n = 50~~$, then $~~R_{50} = 1.999342$
When $~~n = 100~~$, then $~~R_{100} = 1.999836$
We can see that the numbers appear to be approaching 2.
Work Step by Step
We can compute the right Riemann sum for the function using the right endpoint of each subinterval.
$\Delta x = \frac{b-a}{n} = \frac{\pi-0}{n} = \frac{\pi}{n}$
$x_i = \frac{i\pi}{n}$
$R_n = \sum_{i=1}^{n}f(x_i)\Delta x$
$R_n = \sum_{i=1}^{n}(sin~\frac{i\pi}{n})~(\frac{\pi}{n})$
When $~~n = 5~~$, then $~~R_5 = 1.933766$
When $~~n = 10~~$, then $~~R_{10} = 1.983524$
When $~~n = 50~~$, then $~~R_{50} = 1.999342$
When $~~n = 100~~$, then $~~R_{100} = 1.999836$
We can see that the numbers appear to be approaching 2.
