Integral Calculus Study Notes for Civil Engineering

1. INTRODUCTION

Integration is the reverse process of differentiation. It is sometimes called anti-differentiation. The topic of integration can be approached in several different ways. Perhaps the simplest way of introducing it is to think of it as differentiation in reverse.

 1.1 Differentiation in Reverse (Anti-Derivative)

Suppose we differentiate the function F(x) = 3x² +7x-2.  We obtain its derivative as.

This process is illustrated in Figure 1.

                                                                                      Figure 1

In this case, we can say that the derivative of  F(x) = 3x² +7x-2. is equal to 6x+7 . However, there are many other functions which also have a derivative 6x+7. Some of these are  3x² +7x+3, 3x² +7x,3x² +7x-11  and so on. The reason why all of these functions have the same derivative is that the constant term disappears during differentiation. So, all of these are anti-derivatives of . Given any anti-derivative of f(x) , all others can be obtained by simply adding a different constant. In other words.

if F() is an anti-derivative of , then so too  is F(x)+C  for any constant  and this actually describes the definition of Indefinite Integration.

1.2 INDEFINITE INTEGRATION

We call the set of all anti-derivatives of a function as the indefinite integral of the function. The indefinite integral of the function f(x)   is written as

and read as "the indefinite integral of  f(x) with respect to x".  The function f(x) that is being integrated is called the integrand, and the variable  x is called the variable of integration and the C is called the constant of integration.

1.3 Properties of the Indefinite Integral

Basically, there are three properties of anti-derivatives which been applied in order to solve the integration for any kind of functions.

1.4  Integral of Polynomial Functions

Properties of the Integral of Polynomial Functions

Example: 

1.5 Integral of Exponential Functions

Formula of the Integral of Exponential Functions

Example:

1.6 Integral of Logarithmic Functions

Formula of Integral of Logarithmic Functions

1.7  DEFINITE INTEGRATION

In this section, the concept of a “definite integrals” is introduced which will link the concept of area to other important concepts such as length, volume, density, probability, and work.

Based on the Figure, the curve f(x) s nonnegative and continuous on an interval [a,b]. The area of which is under A the graph of f(x)  over the interval [a,b] can be represented by the definite integral. 

Note that there is no constant in definite integral, therefore definite integral is always in number. This is because the constant c is eliminated as shown below.

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