Comments on: How to Add 1 through 100 using Calculus https://betterexplained.com/articles/how-to-add-1-to-100-using-calculus/ Math lessons that click Wed, 27 Dec 2017 20:08:08 +0000 hourly 1 By: Matt https://betterexplained.com/articles/how-to-add-1-to-100-using-calculus/#comment-336297 Wed, 27 Dec 2017 20:08:08 +0000 https://betterexplained.com/?p=10127#comment-336297 I not too long ago thought about this exact same problem, oddly enough. Instead of integrating on the top-right point on each rectangle-bar, integrate on the top-middle point of each rectangle-bar.

To "fix" the integral, instead of integrating x from 0 to x, you integrate from 0.5 to n + 0.5, and you get the mid-point of each bar. In doing some more work, it turns out you can get the exact answer using an integral (and probably an even nicer way to derive the formula for 1 + 2 + 3 + ... + n). For this, you simply integrate x from 0.5 to n + 0.5, and you get the formula (n)(n+1)/2.

Going beyond this seems to be trickier. The formula is 1^2 + 2^2 + 3^2 + ... = (1/3)n^3 + (1/2)n^2 + (1/6)n. If I have the integral from 0.5 to n + 0.5 of x^2 - 1/12, I get the formula. I tested it on the case of n = 5 and got 55 both ways.

]]> By: kalid https://betterexplained.com/articles/how-to-add-1-to-100-using-calculus/#comment-334717 Mon, 20 Mar 2017 19:24:05 +0000 https://betterexplained.com/?p=10127#comment-334717 In reply to Diana.

Thanks Diana, hope it helps them!

]]> By: Diana https://betterexplained.com/articles/how-to-add-1-to-100-using-calculus/#comment-334716 Mon, 20 Mar 2017 18:22:21 +0000 https://betterexplained.com/?p=10127#comment-334716 Thx Khalid... give me some ideas to explain integral in a more interesting way to my students

]]> By: kalid https://betterexplained.com/articles/how-to-add-1-to-100-using-calculus/#comment-334707 Sun, 19 Mar 2017 03:43:42 +0000 https://betterexplained.com/?p=10127#comment-334707 In reply to Rodolfo.

Neat extension! We can account for the extra area by making the sum a bit larger.

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