Computing Darboux Sums

Question

Let $f : [0,1] → R$ be defined by $f(x) := x^3$ and let $P :=$ {$0,0.1,0.4,1$}.
Compute $L(P, f)$ and $U ( P, f )$ . (Notation meaning the upper and lower Darboux Sums)

Answer 1

I don't see how what you've typed is a partition but my answer below applies in general (perhaps the $P$ you've written down is the set of differences so the set of partition points is $\{0,1,5,6\}$ but then this isn't a partition of $[0,1]$; rather it's a partition of $[0,6]$ ...):

The function $f:[0,1]\to \mathbb{R}$ is monotonically increasing. Prove that for any such monotonically increasing function, and any partition $P=0=x_0<\cdots<x_n=1$, we have:

$L(P,f)=\sum_{i=0}^{n-1} f(x_i)(x_{i+1}-x_i)$

$U(P,f)=\sum_{i=1}^{n} f(x_i)(x_{i}-x_{i-1})$.

I hope this helps!