How do we add up infinitely many infinitesimal areas? This is elementary if we know a function of which is the first derivative. If then and
All we have to do is to add up the infinitesimal amounts by which increases as increases from to and this is simply the difference between and
A function of which is the first derivative is called an integral or antiderivative of Because the integral of is determined only up to a constant, it is also known as indefinite integral of Note that wherever is negative, the area between its graph and the axis counts as negative.
How do we calculate the integral if we don't know any antiderivative of the integrand? Generally we look up a table of integrals. Doing it ourselves calls for a significant amount of skill. As an illustration, let us do the Gaussian integral
For this integral someone has discovered the following trick. (The trouble is that different integrals generally require different tricks.) Start with the square of :
This is an integral over the plane. Instead of dividing this plane into infinitesimal rectangles we may divide it into concentric rings of radius and infinitesimal width Since the area of such a ring is we have that
Now there is only one integration to be done. Next we make use of the fact that hence and we introduce the variable :
Since we know that the antiderivative of is we also know that
Believe it or not, a significant fraction of the literature in theoretical physics concerns variations and elaborations of this basic Gaussian integral.
One variation is obtained by substituting for :
Another variation is obtained by thinking of both sides of this equation as functions of and differentiating them with respect to The result is