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 vital to the knowledge of calculus. And to answer the question “What are the uses of integration?” then we can say integration can be used to find areas under the curve surface. According to M. Bourne (2011), one of the first uses of integration was in finding the volume of wine-casks –that have a curved surface. Other uses of integration will include finding the centers of mass, displacement and velocity, fluid flow, modeling the behavior of objects under influence or stress, and so on. In the real world, the knowledge of integration was borrowed to design buildings. For example, in Kuala Lumpur, integration was used to design PETRONAS Towers for strength because it experienced high forces due to wind. Another example is Sydney Opera House that has a very unusual design based on slices out of a ball –different equations (a branch of integration) were employed in building the house (M. Bourne, 2011). Now what is indefinite integral? We know that the process of finding a function f(x) from one of its known values and derivative f ‘(x) or d/dx(x) has two steps. One of these two steps is to determine the formula that gives all the functions that could have f as a derivative, and these functions are called antiderivatives of f while the formula that gives them all is termed as indefinite integral (George, T. 2001). In mathematical definition, a function F(x) is an ant derivative of a function f(x) if F’(x) = f(x) for all x in the domain of f. the set of all antiderivatives of f is the indefinite integral of f with respect to x, symbolized as ʃ f(x) dx. The symbol ʃ is an integral sign which denotes sum. The function f is the integrand of the integral and x is the variable of integration. Indefinite integral results in a function when it’s calculated. An indefinite integral of a function f(x) is also known as the antiderivative of f –as explained earlier. A function F is an antiderivative of f on an interval I, if F'(x) = f(x) for all x in I. From its mathematical...
Please join StudyMode to read the full document