Integral Calculus

The two main operations in the field of calculus are differentiation and integration. Integral calculus holds an important place in calculus which is represented with the formula

Where f is the function of the real variable x with the integral [a,b] of the real line. 

The integral term is also referred to mean the antiderivative where the indefinite integral and the definite integrals. Isaac Newton and Gottfried Leibniz have formulated the principles of integration during the late 17^th century. Integration is connected with the differentiation through   the basic theorem of calculus which has been developed independently where f is the continuous real-valued function that is mentioned on a closed interval [a,b].

Once an antiderivative F of ƒ is known, the definite integral of ƒ over that interval is given by

= F(b) - F(a)

Basic tools:

The basic tools of calculus are integrals and derivatives which has got numerous applications in engineering and science. The integral has been thought to be the infinite sum of rectangles of the infinitesimal width by the founders of the calculus. Bernhard Riemann has given the rigorous mathematical definition for integral. The area of the curvilinear region has been approximated by the limiting procedure through breaking the region into several thin vertical slabs. More number of sophisticated notions in the field of integral has begun to appear from the beginning of the nineteenth century. The functions of two or more variables will be defined by the line integral and a curve which connects the two different points in the space or on the plane replace the interval of integration [a,b]. A piece of surface which is present in the three dimensional space, replaces the curve in a surfaced integral. In the modern differential geometry the integrals of differential forms play a basic role.

Contribution of physics:

The demands and the need that are formed in physics has arose the generalizations of the integral calculus. These integrals of the differential forms play a prominent role in the formation of various physical laws especially the laws of the electrodynamics. Lebsgue integration is one important mathematical theory which is developed by Henri Lebesgue. Of course, there are various modern concepts of integration which were generated during this period.

History:

Reckoning the history of integral calculus will take one back to the ancient Egypt (1800 BC) when Moscow Mathematical Papyrus has demonstrated the knowledge of a formula for a particular volume of the pyramidal frustum. The method of exhaustion is the first ever documented systematic method which is capable of determining integrals.

Questions:

• What is integral calculus?
• What is method of exhaustion?