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I was wondering if there was any process to take the sum of any given functions outputs akin to a integrals to sum the area under the curve. For example, if my function is $$ \frac { \left( x - \frac 1 2 \right) ^ 2 } 2 \text , $$ and I want to find the area under the curve from $ x = 1 $ to $ x = 10 $. I would integrate from $ 1 $ to $ 10 $.

What if I want the sum of all $ Y $ values corresponding to $ x = 1 $ to $ x = 10 $ of the function without computing each separately? Is there anyway I can reduce this operation into one problem instead of plugging in $ 10 $ different values?

Mohsen Shahriari
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AaronSzcz
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1 Answers1

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You could say $$\sum\limits_{x=1}^{10} \frac { \left( x - \frac 1 2 \right) ^ 2 } 2 = \frac18 { \sum\limits_{x=1}^{10}\left( 2x - 1 \right) ^ 2 } = \frac18 { \sum\limits_{m=1}^{20}m ^ 2 } - \frac18 { \sum\limits_{m=1}^{10}(2m) ^ 2 }= \frac18 { \sum\limits_{m=1}^{20}m ^ 2 } - \frac12 { \sum\limits_{m=1}^{10}m ^ 2 } $$

It then helps if you know that $\sum\limits_{m=1}^n m^2 = \frac16 n(n+1)(2n+1)$

Henry
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