Integration
The first application of integration was to measure area. The measurement of
the area under a curve is done by dividing it into approximate rectangular strips,
each with width dx. By measuring the area of each and adding them up we get
the ‘sum’ and so the total area. The notation S standing for sum was introduced
by Leibniz in an elongated form ∫. The area of each of the rectangular strips is
udx, so the area A under the curve from 0 to x is
If the curve we’re looking at is u = x^2 , the area is found by drawing narrow
rectangular strips under the curve, adding them up to calculate the approximate
area, and applying a limiting process to their widths to gain the exact area. This
answer gives the area
A = x^3 /3
For different curves (and so other expressions for u) we could still calculate
the integral. Like the derivative, there is a regular pattern for the integral of
powers of x. The integral is formed by dividing by the ‘previous power +1’ and
adding 1 to it to make the new power.
The star result
If we differentiate the integral A = x^3 /3 we actually get the original u = x^2. If