Subject: Calculus

Integral Calculus

In the previous section, we have just discussed about derivative (Differential Calculus) which is one of the primary subject of Calculus. The other one is the integral or the Integral Calculus. Traditionally, integral and differential calculus are conventionally viewed and treated as separate subjects of Calculus and that no interconnection is seen between the two Calculus subjects.

History


Newton and Leibniz : Great Founders of Calculus

Differential calculus was based on the concept of tangent lines and study of rates of change and infinitesimals which was separately introduced and discovered by Euclid (tangent lines), Archimedes and the Indian mathematics (infinitisemals and rates of change). Moreover, the idea of integral calculus has first sprouted way back 1800 BC when Moscow Mathematical Papyrus first demonstrated the Ancient Egypt's knowledge of a formula for the volume of a pyramidal frustum. Such imminent diversity with regards to history and principles underlying derivative and integral lead people long ago that indeed integral and derivative has no connection.

Derivative and Integral

That idea has changed when Newton and Leibniz saw the connection between derivative and integral. The intimate relationship among these two facets of the Calculus was independently discovered by Newton and Leibniz in their approach to resolve questions relating to their respective works. The truth is without discovering Calculus, Newton could have been unable to create his laws of motions and his theory of universal gravitation.


Newton Laws of Motion

Integral Calculus

Integral calculus is a mathematical entity which could be viewed as an area or a generalization of area. Some common terms for integral are antiderivative and primitive. Bernhard Riemann gives a thorough mathematical concept of the integral. He stresses that integrals are structured using a limiting process which approximates the area of a curvilinear region by breaking down that region into thin vertical slabs. This concept by Reimann is called Reimann Integral. Aside from Reimann integral, there are also other types of integrals such as line and surface integrals.

Line integral,as mentioned in Wikipedia, is defined for functions of two or three variables where the interval of integration [a, b] is substituted by a certain curve or contour connecting two points on the plane or in the space. Line integral is also known as contour integral. While line integral uses line to replace a certain curve, surface integrals uses surface as a replcaement for cure in a three-dimensional space. While Reimann is the most dominant of integrals, line and surface integrals are very useful in differential geometry and physics which applciations are very imminent in electrodynamics.

More Contents Under Differential Calculus
  1. Antiderivative, Indefinite integral
  2. Simplest rules
  3. Sum rule in integration
  4. Constant factor rule in integration
  5. Linearity of integration
  6. Arbitrary constant of integration
  7. Fundamental theorem of calculus
  8. Integration by parts
  9. Inverse chain rule method
  10. Substitution rule
  11. Differentiation under the integral sign
  12. Trigonometric substitution
  13. Partial fractions in integration
  14. Quadratic integral
  15. Trapezium rule
  16. Integral of secant cubed
  17. Arclength