For an ordinary integral of a real function over an interval of the real line, the fundamental theorem of calculus is the statement that the definite integral is an anti-derivative. This means that differentiation undoes integration and vice versa, to the extent that it can.
This has two manifestations: if you differentiate a function f and then integrate back, you get the difference of the function between the endpoints of integration
This formula means that we can use all the methods available for finding anti-derivatives in evaluating definite integrals.
The second is less commonly applied but still useful: if you integrate a function f and then differentiate the integral with respect to its upper endpoint (y above) you get f back again
These properties hold in essentially the same way for the other two kinds of one dimensional integral that we introduced in the last chapter.
And analogues of these statements hold for every kind of integral defined there. We will discuss these in turn.
We begin by providing proofs of these statements in the case of ordinary integrals.
To prove the first claim above we observe that if we divide the interval from a to y into tiny subintervals the result claimed follows if it is true in each subinterval.
In any interval the mean value theorem tells us that the difference in f between its endpoints is their separation times the derivative of f at some intermediate point.
Thus the actual difference in f over the interval from x to x + d, f(x + d) - f(x), can be considered the contribution from that subinterval to some Riemann sum, for some point x' in the interval.
Since the existence of the integral implies that all the Riemann sums converge to it, the particular subset of them obtainable by applying the mean value theorem on each sub-interval must do so as the maximum ds goes to zero, and we find that the integral of the derivative is the change in f between the endpoints in each infinitesimal subinterval.
The sum of these changes is the change in f over the whole interval, which is our integral.
The second claim can be rewritten as the statement
in the limit as dy goes to zero.
Since the integral represents the area of the rectangle with sides dy and f(y) as dy goes to zero, this result holds when f is continuous at argument y.
In that case the value of f for arguments sufficiently close to y are arbitrarily close to f(y) and the area of a sufficiently narrow rectangle will be arbitrarily close to f(y)dy.
To repeat, the implication of these statements is that all the methods we found for anti-differentiation can be applied to determining areas under curves, and more generally, to evaluating definite integrals.