Fundamental Theorem of Calculus - Geometric Intuition

Geometric Intuition

For a continuous function y = f(x) whose graph is plotted as a curve, each value of x has a corresponding area function A(x), representing the area beneath the curve between 0 and x. The function A(x) may not be known, but it is given that it represents the area under the curve.

The area under the curve between x and x + h could be computed by finding the area between 0 and x + h, then subtracting the area between 0 and x. In other words, the area of this “sliver” would be A(x + h) − A(x).

There is another way to estimate the area of this same sliver. h is multiplied by f(x) to find the area of a rectangle that is approximately the same size as this sliver. It is intuitive that the approximation improves as h becomes smaller.

At this point, it is true A(x + h) − A(x) is approximately equal to f(x)·h. In other words,

with this approximation becoming an equality as h approaches 0 in the limit.

When both sides of the equation are divided by h:

As h approaches 0, it can be seen that the right hand side of this equation is simply the derivative A′(x) of the area function A(x). The left-hand side of the equation simply remains f(x), since no h is present.

It can thus be shown, in an informal way, that f(x) = A′(x). That is, the derivative of the area function A(x) is the original function f(x); or, the area function is simply an antiderivative of the original function.

Computing the derivative of a function and “finding the area” under its curve are "opposite" operations. This is the crux of the Fundamental Theorem of Calculus. Most of the theorem's proof is devoted to showing that the area function A(x) exists in the first place, under the right conditions.