Application of Antiderivative





Application of Antiderivative

Antiderivates can be defined as the inverse function of derivatives. An antiderivative of a function f(x) is a function whose derivative is f(x).

Some of the important formulas of Antiderivatives are as follows:-

 

(i)Let f (x) be function of x, then definite integral g of f (x) with respect to x between the limit a & b is devoted by  \displaystyle{\int ^b_a f(x) dx} and defined by  \displaystyle{\int ^b_a f(x) dx \\[3mm] = \lim_{h \to 0} h \sum^n_{r = 1} f(a + rh) \, \, or \, \, \int ^b_a f(x) dx}
= \displaystyle{\lim_{h \to 0}h [f(a + h) + f(a + 2h) + \cdots + f(a +nh)]}  which also known as limit of a sum.

 

(ii)The area bounded by the curve y = f (x), x-axis and x = a and x = b is given by \displaystyle{\int ^b_a f(x) dx \, \, or \int ^b_a y dx}

(iii)The area bounded by the curve x = f (y), y-axis and y = a & y = b is \displaystyle{\int ^b_a f(y) dy \, \, or \int ^b_a x dy}

(iv)Area between two curves y_1 = f_1(x) \, \, and \, \, y_2 = f_2(x) and x = a & x = b is given by \displaystyle{\int ^b_a[f_1(x)- f_2(x)] dx \, \, or \int ^b_a(y_1- y_2)dx} if the graphical about both axes then,  total \, \, area = 4 \times Area \, \, of \, \, AOB

 



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