We partition the interval $[a,b]$ into $n$ equal parts: $$x_k= a+k h$$ for $k=1,2\ldots n$ with $$h=\frac{b-a}{n}\tiny.$$ For the choice of tags, we distinguish the following three cases: choose for each sub-interval $[x_{k-1},x_k]$ always the

1. left endpoint, $\;\,\,s_k=a+(k-1)h\;\;\;$ (left Riemann sum)
2. right endpoint, $\;\!s_k=a+k h\qquad\;\;\;$ (right Riemann sum)
3. midpoint, $\,s_k=a+(k-\tfrac{1}{2})h\;\;$ (midpoint Riemann sum)

The Riemann sums can now be written as follows:

1. left Riemann sum = $\displaystyle h\sum_{k=1}^{n}f\bigl(a+(k-1) h\bigr)$
2. right Riemann sum = $\displaystyle h\sum_{k=1}^{n}f\bigl(a+k h\bigr)$
3. midpoint Riemann sum = $\displaystyle h\sum_{k=1}^{n}f\bigl(a+(k-\tfrac{1}{2}) h\bigr)$

These sums are used to approximate the area under the curve $f$. A visualisation of the Riemann sums is made available below for you to play with, so that you can get a better idea of the differing situations.

Programming task

Write a function Riemannsom(f,a,b,n,methode'left endpoint') that calculates not only the left Riemann sum of the function $f$ on the interval $[a,b]$ when divided into $n$ subintervals, but also the right Riemann sum and midpoint Riemann sum, respectively with the option method='right endpoint' or method='midpoint'.

Apply your function with $n=100$ in the following two cases:

• $\displaystyle \int_0^1\frac{4}{x^2+1}\,\dd x=\pi$
  
  
• $\displaystyle \int_0^{\pi}\sin(x)\,\dd x=2$