Lesson 7 Integration

In the Seventeenth Century, the idea of integration was discovered by both Leibniz and Newton. However, Newton took complete credit for this discovery and used it to find the area of curved figures. Newton took the first steps to define calculus as a mechanical calculator of both theory and practice. In ancient time periods, Greeks did not have the luxury of utilizing calculus. Instead, Greeks would perform the method of exhaustion. This method divided curved figures into a plethora of simple polygons in order to find the area of curved figures. In fact, this is how the value of pi was discovered as 3.1415... Kepler was able to discover the area and volume of ninety-two new curved shapes using this method. Below is a polygon that is divided into smaller polygons and a circle divided into smaller polygons by the method of exhaustion:

However, what if the area of the space between the curve y = x^2 for all values [0, t] and the x-axis needed to be found? The area can still be divided into rectangles and added together in order to find the area. Leibniz developed a notation for this process using the sumation symbol, therefore:

         t                                       t

summation [f(x) delta x] = integral [f(x) dx]

         a                                       a

Using integration directly the area from 0 to t can be caculated more efficiently. And once calulated, this area would be equal to the integral of y=x^2 at t subtracted by the integral of y=x^2 at 0.

 Once observed, Integration and Differentiation are inverse processes:

f(x)

            t

integral [f(x) dx]

            0

               t

d integral [f(x) dx]

dt            0

                                                                                                   t

This is the First Fundamental Theorem of Calculus:   d integral [f(x) dx]=f(t)

                                                                                      dt            0

When dealing with area (as displayed above), the area between the curve and the x-axis on the interval [t,a], is equivalent to the expression below:

                      t

A(t) = integral dA dx + A(a)

                      a dx

This is the Second Fundamental Theorem of Calculus. Here is a video describing the use of the Second Fundamental Theorem of Calculus:

Integration is not only used to find areas of curved figures but is also used in mechanics. And identical to derviatives, acceleration, velocity, and displacement are all related by their integrals.