Motivation for the Fundamental Theorem of Calculus

Graphed below is the function f(x) = x and the (signed) area under the curve between 0 and x, which we called the definite integral of f(t)dt from 0 to x.

Drag the point x around to trace out a new function. How does the new function relate to f?

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Jason McCullough, April 5, 2010, Created with GeoGebra