Definite and Indefinte Integral
Before we can discuss both definite and indefinite integrals one must have sufficient and perfect understanding of the word integral or integration. So the questions that arise from this will be “what is integral or integration?”, “why do we need to know or study integral or integration?” and if we understand its concept then “what are its purposes’? These questions should be answered clearly to give a clear, precise meaning and explanation to definite and indefinite integrals.
To answer the first question in a very plain language, integration is simply the reveres of differentiation. And differentiation is, briefly, the measurement of rate of change between two variables, for example, x and y. This mathematical method can be used to reverse derivative back to its original form. For some one that is familiar with derivative, we know that d/dx (x2) = 2x or in mathematical notation we can write it as f ’(x2) = 2x. This is calculated simply by using the derivative formula nxn-1 where x2 will be 2* x2-1 = 2x.
Now to reverse this derivative we have to use law of integral (power rule) that states for f(x), x = xn+1n+1 (normally written as xn+1n+1 + k) now f(x) = 2x will now be equal to 2 * x1+11+1 = 2* x22 + c = x2
This method of reversing the derivative of a function f back to its original form is what is meant by integral. It is also known as anti-derivative simply because it acts as the opposite of differentiation. Hence, for f (x) = x2 (then d/dx (x2) = 2x) will be equal to ʃ 2x dx. This means f(x) = x2 = f ‘(x) = 2x = f(x) = ʃ 2x dx = x2
The concept of integration helps mathematicians to know the direct relationship between two variables that used to calculate the rate of change. For example, we may know the velocity of an object dropping from the sky or any given height –using the knowledge of derivative, but to know the position of that particular object at a given time we must employ the knowledge of integration. Integration is very valid and
To answer the first question in a very plain language, integration is simply the reveres of differentiation. And differentiation is, briefly, the measurement of rate of change between two variables, for example, x and y. This mathematical method can be used to reverse derivative back to its original form. For some one that is familiar with derivative, we know that d/dx (x2) = 2x or in mathematical notation we can write it as f ’(x2) = 2x. This is calculated simply by using the derivative formula nxn-1 where x2 will be 2* x2-1 = 2x.
Now to reverse this derivative we have to use law of integral (power rule) that states for f(x), x = xn+1n+1 (normally written as xn+1n+1 + k) now f(x) = 2x will now be equal to 2 * x1+11+1 = 2* x22 + c = x2
This method of reversing the derivative of a function f back to its original form is what is meant by integral. It is also known as anti-derivative simply because it acts as the opposite of differentiation. Hence, for f (x) = x2 (then d/dx (x2) = 2x) will be equal to ʃ 2x dx. This means f(x) = x2 = f ‘(x) = 2x = f(x) = ʃ 2x dx = x2
The concept of integration helps mathematicians to know the direct relationship between two variables that used to calculate the rate of change. For example, we may know the velocity of an object dropping from the sky or any given height –using the knowledge of derivative, but to know the position of that particular object at a given time we must employ the knowledge of integration. Integration is very valid and