Infinitesimal Calculus – Integral Calculus
Originally, integral calculus was developed to calculate the surface area underneath the curve of a function. In general, integral calculus is a method to calculate the sum of infinite elements of a function within a continuous space.
Using integral calculus, an infinitely large number of infinitely small areas are added to calculate the area under a graph.
Many narrow strips
A method of integration was first developed by Archimedes during ancient times, and later generalized by Leibniz and Newton. The area under the curve is divided into many rectangles of equal width. The area of each rectangle can be calculated simply by multiplying two measurements, its idth Ax and its height f(x1): A1 = Ax- f(x1).
The area of the curve is found by adding together the areas of the rectangles under the curve and factoring out Ax: A=Ax(f(x1) + f(x2) + …). The narrower the rectangles are (the smaller Axis), the closer the sum is to the actual area under the curve.
Primitive and definite integral
Leibnitz and Newton presented the idea that the derivative of the integral of a function F will lead back to the original function f: F'(x) = dF(x)/dx = f(x). F is said to be the indefinite or primitive integral of the function f. This rule connects the two branches of infinitesimal calculus. The definite integral of f from a to b is: j ” f(x)dx =F(b) – F(a).
This is the limit of the summation of the areas f(x)-dx, from x = a to x = b, as dx becomes infinitely small. The symbol of integration is a long drawn-out “S” from the Latin word summa.
Applications
In physics, when a force is constant, simply multiplying the force by the distance gives the mechanical work done on an object by a force acting over a distance.
When the force varies over the distance, calculus must be used W= , \ s2 F(s) ds.
FOCUSING LIGHT WITH GLASS MICROSPHERES
Transparent spheres with a diameter of just a few light wavelengths can act as lenses, focusing light. They also have a tendency to arrange themselves on a surface in a single compact layer. Thus, a layer of microspheres can focus the light of a single laser pulse to produce millions of similar structures, only a few dozens of nanometers in size.
In addition to processing materials, microspheres play an important role in semiconductor technology and micro and nanomechanics. To take full advantage of their focusing
properties, a special concept from calculus must be used— the Bessoid integral—which takes the wave nature of light into account.
INTEGRAL CALCULUS AND VOLUME
A practical application of integral calculus is the determination of the volume of three-dimensional figures formed by rotating a function !(x) around the x or y axis. A cone, for instance, is formed through the rotation of a linear function, l(x)=ax (a line that passes through the origin). To calculate its volume, we can divide the cone into extremely thin slices, which when added together approach the cone’s volume The thinner the slices (differentials, or dx), the more exact the approximation.
As dx approaches zero, the sum of the slices will exactly equal the volume of the cone. The sum of this infinite number of slices is an integral, or a linear depiction which assigns a numeric value or function to a given domain of the integration.