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Integral from (-a) to a of x^3 is 0. But the *area* is 2*integral of 0 to a of x^3.

See?

]]>Derivative-as-angle is a bad idea to start with. It doesn’t even generalize properly to a two-dimensional domain. It should have long ago been completely replaced by the much more general (and still easy to understand) local-approximation-by-a-linear-function.

However, integral-as-area generalizes quite well, even to the most abstract setting (measurable function on a measure space). What doesn’t generalize is one concrete and very simple-minded method of /calculating/ the integral (the so-called ‘Riemann-Integral’).

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