Integral Calculus

[Audio] Integral Calculus". Good afternoon and welcome. Today we're going to be talking about integral calculus, a branch of mathematics which deals with the calculation of the area of irregular regions and the solving of problems involving antiderivation and integration. Let's get started..

Introduction Integral calculus is a type of calculus of mathematics that studies two linear operators that are related and is used for calculations involving arc length, pressure, area, center of mass, volume, and work. In many science and technology programs, Calculus is a.mong the first courses taught, allowing students to study and model real problems in ways that can be applied to their professional lives. However, instructors often emphasize the application of techniques, rather than the acquisition of notions that are directly relevant to engineering. This can lead to students failing Calculus and abandoning their professional ambitions. (Ellis, Kelton, & Rasrnussen, 2014). Christensen (2008) has pointed out that it can be difficult to connect the abstract forrnalisrn of mathematics with the necessary applicable skills in each profession. Calculus is one of the fundamental cou.rses in mathematics and has been introduced into the secondary school cuniculu.rn in Malaysia as one of the options in forrn four and forrn five mathematics. It is important for students taking engineering and science to excel in calculus, including integral calculus, as a growing body of research has shown that students have difficulties understanding the concept of integral calculus..

[Audio] To begin, indefinite integral calculus is the process of evaluating an integral as a function of the variables it contains. This is done by finding the anti-derivative of the given integral and then evaluating it. After that, definite integral calculus is used to determine the area under the curve described by the integral. This is done by finding the definite integral and then applying the fundamental theorem of calculus. Finally, definite integrals can be used to solve problems involving the area of irregular regions. This involves using the calculus of areas and finding the definite integral for each region to determine the total area..

[Audio] Students should gain the knowledge and skills to solve any problem related to integral calculus by the end of this module. This includes being able to identify the indefinite antiderivative of a function, calculate the definite antiderivative of a function, and determine the area bounded between a given function and the x-axis. To successfully answer questions related to integral calculus, it is necessary that students understand and are able to apply these concepts..

[Audio] Integral calculus is a branch of mathematics that revolves around the idea of finding the area between curves, or the regions enclosed by functions. It is useful for computing the area that is difficult or impossible to calculate with conventional methods. The key operations of integral calculus are antiderivation and integration. Antiderivation helps to find the primitive of a function which enables us to find the area under a given curve. Integration, on the other hand, is the opposite of differentiation and helps us to find the area under a given curve or surface. By using these two operations, we can calculate areas which are not easy to calculate with conventional methods. This makes integral calculus a very powerful tool for solving different kinds of problems. We hope this slide has provided you with a better understanding of the fundamentals of integral calculus..

[Audio] Antiderivatives, also known as indefinite integrals, are a fundamental operation in calculus. They are the reverse of differentiation, as the derivative of a function gives us the rate of change of the function, the antiderivative gives us the area under the function. To determine the indefinite antiderivatives of a given function, let F(x) be the antiderivative of f(x). This means that the indefinite integral of f(x), denoted by ∫ f(x)dx, is equal to F(x) + C, where C is the constant of integration. In this notation, the integral sign ∫ is an elongated S shape and the dx indicates that the function is to be integrated with respect to variable x. As such, to find the antiderivative of a given function, we must determine the area over which the function is defined and then calculate the area..

[Audio] The theorem of integration states that for any function f(x), the integral of f(x) with respect to the variable x is x plus C. Taking the example of the integral of dy, the result is y plus C. This pattern can be seen with other functions like a, b, f, r, m, p and z, where the result of the integral of each of these functions is the respective variable plus C. This theorem and its applications will be discussed in this slide..

[Audio] Theorem 2 states that when a constant is behind the integral function, the integral can be simplified to ax + C. For example, when we have ∫7dx, we move the constant 7 behind the integral and apply Theorem 2 to simplify it to 7x + c. Similarly, we can do the same for ∫ 8dx, 8∫dx, and -3∫dx respectively to get 8x + C, -3x + C and πx + C. This theorem is very useful when evaluating integrals..