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Animations unifying the relationships among rates of change and accumulations of power functions
$x^3 \leftrightarrow 3x^2$ --> click here

$x^2 \leftrightarrow 2x$click here

The formulas illustrated are:<

$(n+1)^2=1+ 2 \sum_{j=1}^n j + 1 \sum_{j=1}^n 1$

$(n+1)^3=1+ 3 \sum_{j=1}^n j^2 + 3 \sum_{j=1}^n j + 1 \sum_{j=1}^n 1$

which may be rearranged to obtain the familiar summation formulas used in integrating $f(x)=x$ and$g(x)=x^2$ directly:

$\sum_{j=1}^n j = {n (n+1) \over 2}$

$\sum_{j=1}^n j^2 = {n(n+1)(2n-1) \over 6}$

These demonstrations show that the coefficients of $n$ and $1 \over n+1$ in the differentiation and integration formulas share a common source: Pascal's triangle

They foreshadow the deeper connection between rates of change and geometric boundaries made systematic in the Stokes' theorems.

We ultimately wish to emphasize other aspects of the duality between differentiation and integration in a more unified and symmetrical fashion, exhibiting in parallel the linearity rules for both differentiation and integration, the product rule beside integration by parts, the chain rule along with substitution, etc.

Along with summation formulas, discrete analogues of other calculus rules, such as "summation by parts" will also be highlighted.

Andre Cherkaev