How to calculate the upper and lower sum of a definite integral using the darboux definition

Question

I have no idea how to calculate the L(f,P) and U(f,P) for this question. Can anybody please help me out?

$f(x) = -x, x\in[0,1];P=\text{{0,$\frac{1}{8}$,$\frac{1}{3}$,0.9,1}}$

Answer 1

Let’s say $P=\{x_0, x_1 ,...,x_n\}$ then $$L(f,P)=\sum_{i=1}^n m_i (x_i-x_{i-1})$$ Where $m_i$ is the infimum of $f$ on the interval $[x_{i-1},x_i]$ (Which is like the minimum value). For example in your case $m_1 = -1/8$ since that is the minimum value of $f$ on $[0,1/8]$. So your first term in the sum will be $-1/8 \cdot 1/8$, since $1/8$ is the length of the first interval. Do this for the other intervals and add them all.

For the upper sum you do the same thing but use the supremum (maximum) of $f$ on the sub intervals.