I can't quite get the upper and lower sums of $f(x)=x^2$ from $[a,b]$ (where we are defining the function) .
Here I'm talking about Darboux sums of the form $U(f, P_n) = \displaystyle\sum_{i=1}^n M_i(t_i-t_{i-1})$ and $L(f,P_n)=\displaystyle\sum_{i=1}^n m_i(t_i-t_{i-1}) $ where $m_i=inf\{f(x): t_{i-1} \le x \le t_i\}$ and $M_i=sup\{f(x): t_{i-1} \le x \le t_i\}$.
(Also, we assume a partition $P_n=\{t_{\mathbb{0}},t_1,t_2,…t_n\}$)
I was thinking that $t_i= \frac{i(b-a)}{n}$, but if we do this in the end the sum is multiplied by $\frac{(b-a)^3}{n^3}$ which is a bit different from the expected $\frac{b^3-a^3}{n^3}$ that easy way to calculate the integral yields. (where the $\frac {1}{n^3}$ cancels out with the limit of the sum that ends up as $\displaystyle\sum_{i=1}^n=i^2$).
So if anyone could shine some light on how to work this out (pick a better $t_i$ or if I'm doing the sum incorrectly) it would be very much appreciated thanks.
Best Answer
To make things easier, consider $[0,a]$. The function $x^2$ is increasing over $[0,a]$ and it is continuous there. Thus, if we partition $[0,a]$ with $$P=\{t_0,\dots,t_n\}$$ we will have $$M_i=t_i^2$$
and $$m_i=t_{i-1}^2$$
since a continuous functions attains it maximum and minimum in a closed interval. Thus
$$L(f,P)=\sum_{i=1}^n t_{i-1}^2(t_i-t_{i-1})$$
$$U(f,P)=\sum_{i=1}^n t_{i}^2(t_i-t_{i-1})$$
As you say, consdier the partition where $$t_i=\frac{ia}{n}$$
Then $$\eqalign{ & L(f,P) = \sum\limits_{i = 1}^n {{{\left( {i - 1} \right)}^2}\frac{{{a^2}}}{{{n^2}}}} \frac{a}{n} = \frac{{{a^3}}}{{{n^3}}}\sum\limits_{i = 1}^n {{{\left( {i - 1} \right)}^2}} \cr & U(f,P) = \sum\limits_{i = 1}^n {{i^2}\frac{{{a^2}}}{{{n^2}}}} \frac{a}{n} = \frac{{{a^3}}}{{{n^3}}}\sum\limits_{i = 1}^n {{i^2}} \cr} $$
Since $$\sum\limits_{i = 1}^n {{i^2}} = \frac{{n\left( {n + 1} \right)\left( {2n + 1} \right)}}{6}$$ we get $$\eqalign{ & L(f,P) = \frac{{{a^3}}}{{{n^3}}}\sum\limits_{i = 0}^{n - 1} {{i^2}} = \frac{{{a^3}}}{3}\frac{{n\left( {n - 1} \right)\left( {2n - 1} \right)}}{{2{n^3}}} \cr & U(f,P) = \frac{{{a^3}}}{{{n^3}}}\sum\limits_{i = 1}^n {{i^2}} = \frac{{{a^3}}}{3}\frac{{n\left( {n + 1} \right)\left( {2n + 1} \right)}}{{2{n^3}}} \cr} $$
from where $$\eqalign{ & L(f,P) \to \frac{{{a^3}}}{3} \cr & U(f,P) \to \frac{{{a^3}}}{3} \cr} $$
Then $$\int\limits_a^b {{x^2}dx} = \int\limits_0^b {{x^2}dx} - \int\limits_0^a {{x^2}dx} = \frac{{{b^3}}}{3} - \frac{{{a^3}}}{3}$$