The Fundamental Theorem of Calculus
The fundamental theorem of calculus is one of the most important theorems in the history of mathematics. It states that, given an area function Af that sweeps out area under f (t),
the rate at which area is being swept out is equal to the height of the original function. So, because the rate is the derivative, the derivative of the area function equals the original function:
you can also write the above equation as follows:
Break out the smelling salts.
Now, because the derivative of Af (x) is f (x), Af (x) is by definition an antiderivative of f (x). Check out how this works by looking at a simple function, f (t) = 10, and its area function,
According to the fundamental theorem,
Thus Af must be an antiderivative of 10; in other words, Af is a function whose derivative is 10. Because any function of the form 10x + C, where C is a number, has a derivative of 10, the antiderivative of 10 is 10x + C. The particular number C depends on your choice of s, the point where you start sweeping out area. For a particular choice of s, the area function will be the one function (out of all the functions in the family of curves 10x + C) that crosses the x-axis at s. To figure out C, set the antiderivative equal to zero, plug the value of s into x, and solve for C.
For this function with an antiderivative of 10x + C, if you start sweeping out area at, say,
or just 10x. (Note that C does not necessarily equal s. In fact, it usually doesn’t
When s = 0, C often also equals 0, but not for all functions.)
The figure shows why Af (x) = 10x is the correct area function if you start sweeping out area at zero.