Integral Calculus

Integral Calculus.

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..

Overview One of the most important application of integral calculus is calculating the area of irregular regions. Unlike square, rectangle, and circles, these have their own specific to determine their area. In this module, you will be working first evaluating indefinite integral as a function then determining its definite integral leading to understanding and determining the area of plane region.

Learning Objectives At the end of the module, the students should be able to demonstrate mathematical knowledge and skills in solving problems involving Integral Calculus. Specifically, students are expected to: Determine the indefinite antiderivatives of a given function. Determine the definite antiderivative of a given function. Determine the area bounded by the given function and the given x-axis..

BASIC INTEGRAL CALCULUS.

ANTIDIFFERENTATION The process of determining a function where derivative is known. The process of reversing the process of differentiation. Determine the indefinite antiderivatives of a given function Let F(?) be as antiderivative of f(?), then F(?) + ? is the indefinite integral of f(?) denoted by ∫ ?(?)??. That is, ∫f(x)dx=F (x)+C ∫ : an elongated S meaning sum dx: refers to the fact the function f(x) is to be integrated with respect to variable x. C = Constant of integration.

Indefinite Integral (Theorems of Integration) Theorem 1: ∫ dx= x+c We will now have integration with respect to the variable x. Whereas, integral of dx is simply x plus C. For example, integral of dy is the same as the value of y+C . ( ∫ dy = y+c ) Further examples are listed. ∫ da= a+C ∫ db = b+C ∫ df = f+C ∫ dr = r+C ∫ dm= m+C ∫ dp = p+C ∫ dz = z+C.

Theorem 2: ∫ adx =a∫ dx= ax+c , Where “a” is a constant. In this case, theorem number 2, you put your constant behind the integral function and what will be left inside is your integral of dx (a∫ dx ). In theorem number 1 we already know that integral of dx is simply x + C. Thus, we can simplify that integral of adx ( ∫ adx ) is ax + C. For example, ∫7dx. We will now put the constant behind of the integral function which is 7. Then integral of dx is x + C. Hence integral of 7dx ( ∫7dx ) is 7x + c. ∫ 7dx=7 dx → theorem no. 2 applied. = 7x + C → theorem no.1 applied. ∫ 8dx=8∫ dx =8x + C ∫ -3dx=-3∫ dx = -3x + C ∫ πdx=π∫ dx = π x+C.

Theorem 4: ∫ [f(x)±g (x)] dx= ∫ f(x)dx±〗 ∫ g(x) dx In this theorem, getting the integration of a given 2 or more terms or expression, do it separately or part by part. Get the integration of each expression and simplify it at the end. For example, ∫ ( 3x^2+5x-6)dx. In this case, you are given a trinomial. A trinomial is a polynomial with three terms. In here, we have 3x^2,5x and-6 So, following theorem number 4, we are to get the integration of each of the given terms. We can do it by applying the previous theorems discussed..

Figure 1 shows the the change in the value of the integral as x changes from a to b referred to as definite integral. It is also a number defined as the limit of Riemann sum. It is expressed by:.

Examples: Evaluate the following definite integral.

Scientific Calculator Integration. Now, we are going to evaluate the same function using scientific calculator, specifically CASIO fx - 991 ES plus..

Substitution of the upper limit = 1 and lower limit = 0 Simplifying Answer.

Screen output.

Substitution of the upper limit = 1 and lower limit = 0 Simplifying Answer.

Scientific Calculator Integration Input. Screen output.

Substitution of the upper limit = 1 and lower limit = 0 Simplifying Answer.

e00é)eoo o. Screen output.

Determine the area bounded by the given function and the given x-axis. Areas of Plane Region The definite integral represents the area bounded by the graph of the function and the x-axis. However, there are negative (-) definite integral which implies that the area bounded by the graph of the function is below the x-axis. Unit to be used is square units or units squared. For example, we are going to determine the area bounded by y = 2x and the x-axis from 0 to 2..

In this figure (referring to example #1) the red line in this graph represents your y = 2x. The shaded part in the figure that is shown above is what we call the area bounded by that line together with your base (referring to x – axis) which is the limit from 0 to 2 that is equal to 4 square units..

What if the shaded area is below the x-axis?. 4.1 - l)dx f1-1(x2 - l)dx f(x = x2 1[-3 —(—1)1 - l)dx - l)dx -2 - l)dx Solution: 2 1 3 = [L-11— ( 1) 3 2 2 4 The indefinite integral is — — because the shaded graph is below the x-axis. Since we are looking.

What if the shaded areas are above & below the x-axis? Answer/Solution : Area addition Postulate.

Solution: BEFORE f 21 _ (x2 — x)dx x) (x2 — x)dx = (x2 — x)dx = f2_1 (x2 — x)dx = (x2 — x)dx =.

Answer the ff question It is known as the constant of integration. It refers to the fact the function F (x) is to be integrated with respect to variable x. An elongated symbol that means sum. Is it the process of determining a function where derivatives are known..

Key answer 1-4 A B A D.

Key answers to 5-7 C D A.

Key answers to 8-9 C A.

Key answer to #10 B.

Reference Chris Rasmussen et al, (2014). “STUDENTS WHO SWITCH OUT OF CALCULUS AND THE REASONS THEY LEAVE” https://files.eric.ed.gov/fulltext/ED584594.pdf Christensen O. R. (2008). “ Closing the gap between formalism and application — PBL and mathematical skills in engineering. Teaching Mathematics and its Applications, 27(3), 131-139. https://www.researchgate.net/publication/30931162_Closing_the_gap_between_formalism_and _application--PBL_and_mathematical_skills_in_engineering.