Given a function and a set , the integral
is simply the volume under the graph of over the given set.
Assume that we would like to calculate the volume of a pyramid with a base in form of a isosceles right triangle (with boundary of axes and line ) and height , so in point the height is given by the following formula . So the volume is
where is the triangle mentioned above.
How to calculate this integral? It will be described below.
Fubini theorem states that such an integral is the integral over of the integral over , and i the same as integral over of the integral over , in other words (for functions nice enough):
where and are such that
Obviously you have to treat the outer variable as a constant parameter when calculating the inner integral.
Therefore, returning to our example of the pyramid, the section at has area , so the volume is:
which is consistent with our knowledge about geometry.