So I answer a question about the real analysis. I answered the question, but I'm not sure I did it right. Could someone look at my work and let me know if it's going in the right direction?

The question is: by choosing a suitable preparation from $ (1, 3) $ , show that $ int ^ 3_1 2x ^ 2 -1 dx $ exists and finds its value.

So, to prove it exists, I think I have to prove the lower sum $ s_d $ and the upper sum $ s ^ u $ are equal?

I decided for my suitable preparation $ (1,1+ frac {2i} {n}, …, 1+ frac {2i-1} {n}, 3) $

I have the feeling that I chose the preparation incorrectly.

I work from there $ 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 use that to train $ s_d $ and $ s ^ u $ but I get very confusing answers, I was then given the tip $ sum ^ n_ {i = 1} i ^ 2 = frac {n ^ 3} {3} + frac {n ^ 2} {2} + frac {n} {6} $ So I know that I have to get a similar equation.

Any help would be appreciated.