What is meant by Integration?

Answer 1

In Calculus, integration is the process of finding the area under the curve of a function, usually between two boundaries.

For example, the area under the curve of the graph y=x between 0 and 1 (the two boundaries) is equal to the area of the triangle formed by the x axis, the graph and a vertical line at x=1. Since this triangle covers half the area of a square of length 1 unit, the integral of y=x from 0 to 1 is 1/2.

For more complex curves such as y=x^2, integration is easier and more accurate by finding the anti-derivative, or integral, of y=x^2. Finding the anti-derivative, as the name suggests, is the reverse process of finding a function's derivative. So, the anti-derivative of x^2 is the function whose derivative is x^2. I'm assuming you are familiar with differentiation (the process of finding a derivative of a function) if you are doing problems with integration. So, the anti-derivative of x^2 is (1/3)x^3. The purpose of finding the anti-derivative is to use the Fundamental Theorem of Calculus, which states that the area under the curve of a function from a to b (the two boundaries) is equal to the difference between the values of the function's anti-derivative at b and a. So, plugging in 0 and 1 again for x gives:

Area under curve of x^2 from 0 to 1 = (1/3)(1)^3 - (1/3)(0)^3

= (1/3) - 0

= 1/3

Notice the reverse order of the boundaries, since subtraction is not commutative (order matters).