Fundamental Theorem of Calculus I

Consider the definite integral of an arbitrary function that represents the area under the curve.

If the lower limit is fixed, and the upper limit is assigned a variable, the resulting definite integral then behaves as a function of x, g(x) known as the accumulation function. If x changes, the area under the curve also changes.

The derivative of g(x) shows how the area changes, which can be found by returning to the limit definition.

In this g(x + h) − g(x) represents the area of a thin vertical strip under the curve between x and x + h. As this strip becomes infinitely narrow, its area can be approximated by a rectangle with width h and height f(x).

Then its derivative is the function in the integrand, f(x). This is known as the first version of the Fundamental Theorem of Calculus.

For example, in a series RC circuit, the charge present in the capacitor at any given time t is represented by the integral.

The current through the capacitor, the instantaneous rate of change of its charge, is equal to the expression in the integrand by the Fundamental Theorem of Calculus.