So I'm completing a question relating to real analysis. I have answered the question but I am not wholly sure I have done it correct. If someone could look at my working and let me know if it is along the right lines?
The question is: by choosing a suitable dissection of $[1, 3]$ ,show that $\int^3_1 2x^2 -1 dx$ exists and find its value.
So in order to prove it exists I believe I need to prove that the lower sum $s_d$ and the upper sum $s^u$ are equal?
I chose my suitable dissection to be $[1,1+\frac{2i}{n},...,1+\frac{2i-1}{n},3]$
I have a feeling that I have chosen the dissection incorrectly.
From there I am working out $m_i =$ inf $f(\frac{1+2(i-1)}{n}, \frac{1+2i}{n}) $ = $f(\frac{2i-1}{n})$
and $M_i =$ sup $f(\frac{1+2(i-1)}{n}, \frac{1+2i}{n}) $ = $f(\frac{1+2i}{n})$
I would then use this to work out $s_d$ and $ s^u$ but I am getting very confusing answers, I was given then tip that $\sum^n_{i=1} i^2 = \frac{n^3}{3}+ \frac{n^2}{2}+ \frac{n}{6}$ so I know I need to get an equation similar to this.
Any help would be greatly appreciated.
The partition points are $$1,1+2/n, 1+4/n,...,1+2n/n$$ and your function is increasing so the inf happens at left end and sup happens at the right end of your sub intervals.