Definite Integrals
The beauty and importance of the integral calculus is that it provides a systematic way for the exact calculations of many areas, volumes and other quantities.
Consider the region R in the plane. The region R is bounded by the curve y= f(x), x-axis and two vertical lines x = a and x = b, where b > a.
Assume that y = f(x) is a continuous and increasing function on the closed interval [a, b].
Let ∫ y dx = F(x) + c
If x = a, then the area up to x= a, Aa is
∫ y dx = F(a) + c
If x = b, then the area up to x= b, Ab is
∫ y dx= F(b) + c
The required area of the region R is
Ab − Aa = = (F(b) + c) − (F(a) + c)
a∫b y dx = F(b) − F(a)
a∫b f(x) dx = F(b) − F(a)
This integration gives the area of the region R bounded by the curve y = f(x), x axis and between the lines x = a and x = b. a & b are called the lower and upper limits of the integral.