Title UNIT-5 Integral Calculus and Applications.pdf ✅

DEPARTMENT OF COLLEGIATE AND TECHNICAL EDUCATION DEPARTMENT OF SCIENCE GOVERNMENT POLYTECHNIC RAICHUR PREPARED BY: RAMACHANDRA SUTAR 2022 ENGINEERING MATHEMATICS UNIT-5, INTEGRAL CALCULUS AND ITS APPLICATIONS STUDY MATERIAL N E A R G O V T I T I C O L L E G E A M A R K H E D L A Y O U T R A I C H U R - 5 8 4 1 0 3
Unit-5_Integral Calculus_Ramachandra S_GPT Raichur-117 2 UNIT-5, INTEGRAL CALCULUS SYLLABUS: 1. Definition of an indefinite integral. Listing the Integrals of standard functions. (Algebraic, trigonometric, exponential, logarithmic and inverse trigonometric functions) 2. Rules of Integration. Evaluation of integrals with simple integrands and their combinations 3. Rules of Integration. Evaluation of integrals with simple integrands and their combinations. Problems 4. Evaluation of integrals with simple integrands and their combinations. Problems 5. Evaluation of integrals by Substitution method 6. Evaluation of integrals by Integration by parts 7. Evaluation of integrals by Integration by parts. Problems 8. Definition of definite integrals and their evaluation 9. Evaluation of Definite integrals. Problems 10. Area enclosed by the curves by integral method 11. Volume generated by the curve rotated about an axis by integral method 5.0 Introduction: The study of Integral Calculus consists in developing techniques for the determination of integral of a given function. This subject finds extensive applications to Geometry, Natural and Social Sciences. In this unit, we shall be concerned with applications in relation to the determination of Plane areas, Lengths of arcs and Volumes and Surfaces of solids of revolution, Centre of Gravity and Moment of Inertia. Historically, the subject arose in connection with the determination of areas of plane regions and was based on the notion of the limit of a type of a sum when the number of terms in the sum tends to infinity and each term tends to zero. In fact, the name Integral Calculus has its origin in this process of summation and the words 'To integrate' literally means 'To find the sum of. It is only afterwards that it was seen that the subject of that it was Integration can also be viewed from the point of view of the Inverse of differentiation. We shall not develop the subject in the historical order and, as done above, start by defining Integration as the Inverse of differentiation.
Unit-5_Integral Calculus_Ramachandra S_GPT Raichur-117 3 5.1 Indefinite Integral: Let y=f(x), be a function such that F(x) dx dy = .The function F(x) is called the derivative or the differential coefficient of f(x) with respect to x. The function f(x) itself is called an integral or a primitive or an anti derivative of F(x), with respect to x. This is denoted by  F(x)dx = f (x)      − − = + + = = = = = = = =  = dx x x x x x x xdx x x x x xdx x x x x dx 1 2 2 1 2 2 3 3 2 2 / tan 1 1 Thus, 1 1 (tan ) dx d (sin ) cos Thus, cos sin dx d 2 ( ) 2 Thus, dx d 3 ( ) 3 Thus, dx d wkt Thus, F(x)dx f(x) f (x) F(x) This process of finding a function f(x), whose derivative is F(x), is known as integration or anti differentiation. We know that the derivative of a constant c is zero. Thus ( ) ( ) ( )   = + + = + = = = F x dx f x c c F x F x dx d f x dx d f x c dx d ( ) ( ) ( ) ( ( ) 0 ( ) The constant c is called a constant of integration .Further, as the constant ‘c’ is arbitrary the integral (or anti derivative) of a function is not unique. The entire class of integrals of F(x) w.r.t.x is known as indefinite integral of F(x), w.r.t.x In the symbol, F(x)dx , function F(x) is called integrand, the variable ‘x’ is called variable of integration, the symbol  is called integral sign, which is an elongated S, which is the first letter of Summation
Unit-5_Integral Calculus_Ramachandra S_GPT Raichur-117 4 5.1.1 Standard Integrals: We have defined integration as the inverse process of differentiation. Thus using the differential coefficients of basic elementary functions, which we have already seen, we can prepare a list of integrals of certain functions. This list is called the list of elementary standard forms of integrals. ( ) c x dx x c x c x c x dx x dx x dx x c x c x c x c x dx x dx x xdx x c c x c x c x xdx x dx c x c x x dx c x c x x dx c x c x xdx x c c x c x c n x x dx n x x n c n x dxd n n n n n  = − + + = − + − + = − + = =  = + + = + = + − + = =  = + + = + = + + = = + = + + = + = + + = + = + + =  = + + = + + = = = + =  =  + = + =      + +                  − + − − − + − + + + + + + + − + 1 1 1 2 1 1 1 8. 2 1 2 2 1 1 2 1 1 7. 3 2 3 2 2 3 1 2 1 6. 3 1 4 5. 2 1 3 4. 1 1 2 3. dx 0 1 2. dx 1dx x dx Using above result 1 1 1 1 1 1. 2 2 1 1 2 2 2 1 1 2 1 2 1 2 3 2 3 2 3 1 2 1 2 1 3 1 4 3 2 1 3 2 1 1 2 0 1 0 1 1 1 1 ( ) ( )    = − + − + = − − + =  = + + = xdx x c x c x x dx d xdx x c x c x dx d sin cos 10. cos ( sin ) 0 sin cos sin 9. sin cos