In this article first we define Integration, in mathematics it is an important concept. Its inverse definition is also equally important. Integration is one of the main operations from two basic operations of calculus. In simple form we can define that integration means to calculate area. Now we define mathematically, suppose we have a given function (f) with real variable (x) over an interval [a, b] for a given real line, then we expressed this function as ∫f(x) dx. Integration means calculation of area of the region in XY-plane, which is bounded by the graph of function (x). Area above from the X-axis adds the total value and area below the X-axis subtracts from the total value.
The term integrals also known as antiderivatives. Suppose we have a given function is (F) and derivative of this function is (f). In this case it is known as indefinite integral and can be expressed as ∫f(x) dx. The notion of antiderivative are basic tools of calculus. It has many applications in science and field of engineering. Integral is an infinite sum of rectangles of infinite width. Integral is based on limiting procedure of area. Line integral means function with two or three variables where closed interval are replaced by any curve. Curve may be made in any plane or space. In surface integral in place of plane a short piece of surface is used.
There are various integral properties; integral properties are for the definite notion of antiderivative based on the certain theorems. First theorem is, suppose M(x) and N(x) are two defined functions. They are also continuous function in interval [a, b], then we have linearity property for the notion of antiderivative which can be expressed as
∫ [M(x) +N(x)] dx= ∫M(x) dx + ∫N(x) dx
∫a. M(x) dx= a∫M(x) dx. a is an arbitrary constant and we carry out the constant term from the function.
Second theorem is, suppose function f(x) is defined which is continuous in closed interval [a, b], then we have some special property of integral such as…
∫f(x) dx= 0, when limit are same.
∫f(x) dx= ∫f(x) dx+∫f(x) dx, when limits are divided between interval s like a to c and c to b.
∫f(x) dx= -∫f(x) dx, when upper limit becomes lower limit and lower limit becomes upper limit.
There are many integral types such as definite integral in which function is continuous and define in a closed interval. Other types are indefinite integrals, surface integrals, double integral known as Green’s theorem, triple integrals known as Gauss divergence theorem and line integrals
