Indefinite Integral. Day 1. What is Integration If the differential coefficient of a given function F(x) is given by f(x) then

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1 Chapter 1 Indefinite Integral Indefinite Integral Day 1 What is Integration If the differential coefficient of a given function F(x) is given by f(x) then The process of finding the anti derivative of a given function If is called integration The function is called the integrand and the function is called the integral of is known as variable of integration, is the symbol of integration. Now, we know that, Then But (k is a constant) thus Here is an arbitrary constant, which signifies that for a fixed integrand the integral may assume infinite number of values. Hence it is called Indefinite Integral What is Geometrical Significance Form this point of view an indefinite integral is a family of curves, each of which is obtained by translating one of the curves parallel to itself upwards or downwards. Thus in short, What is the Importance of constant C We know that Also From 1 and 2 Page 1

2 Indefinite Integral This is a wrong conclusion. If Constant of integration C is introduced then This is true for only It not only makes the integral general but also makes the two indefinite integral comparable. Fundamental Integration Formulas Based upon definition and various standard differential formulas we achieve the following integration formulas: Page 2

3 Indefinite Integral What are the various Operations on Integration Page 3

4 Indefinite Integral Self Efforts Solution Page 4

5 Indefinite Integral Day 2 Integration by Substitution Theorem (i) Proof In a previous list of formulas, if in place of x we have ax + b, then the same formula is applicable but we must divide by coefficient of x or derivative of (ax + b) i.e. a. For e.g. Theorem (iii) Integrals of the form, where m, n are positive integers. (i) If m is odd i.e. power of is odd. Put. (ii) If n is odd, i.e. power of is odd, Put. (iii) When m and n are both odd positive integers substitute Page 5.

6 Chapter 2 Definite Integral Definite Integral Day 1 What is Definite Integral? Let be a function of x defined in the closed interval [a, b] and for all x in the domain of then Where be another function, such that is the definite integral of function f(x) over the interval [a, b] a = lower limit b = upper limit This is called Newton Leibniz formula & hold for continuous function is the interval. If it is discontinuous at some point in [a, b], then the integral should be separately evaluated for each interval. Page 68

7 Definite Integral Self Efforts Solution Page 69

8 Definite Integral Geometrical Interpretation of Definite Integral is numerically equal to the area bounded by the curves the x-axis and straight lines In general represents algebraic sum of the areas of the figure bounded by. The curves x axis and the straight lines. The areas above x axis are taken with + sign & areas below the x axis are taken with minus sign. If area bounded by is being asked We should realize diff. between Area & Definite Integral. Full Area enclosed by We can see the difference between area & definite integral. Evaluation of Definite integrals by substitution: (1) When the variable in a definite integral is changed, the substitution in terms of new variable should be effected at three Places. Page 70

9 Definite Integral (1) In the integrated (2) In the differential, e.g. dx (3) In the limits Please. note that substitution is not valid if it is not continuous in the interval [a. b] If we consider Now deferential integral cannot be negative. Moreover, substitution x = 1/t is discontinous at t = 0, the substitution. Properties of Definite Integrals. Property I Intrigation in independent of the change of variable Proof : - Let ϕ(x) be a atideriative of f(x) From (1) and (2) Page 71

10 Definite Integral We can see Property II If the limits of a definite integral are inter changed then its value changes by minus sign only Proof: Let ϕ(x)be Anti derivative of f(x). Then Property III Proof: From (1) and (2) Generalization Page 72

11 Area Chapter Definite Integral 3 Day 1 Enquiry How it is possible to find the area enclosed by a curve and x-axis? We know gives the algebraic sum of areas between f(x), x-axis and ordinate x = a & x = b. Consider the strip of width δx and length y from x-axis clearly Area ABMN < Area ABCN < Area ABCD As the area of lower and upper rectangle that to be equal. Thus by Sandwich theorem, Area ABCN = y δx So, Area of is given by Find the area common to the parabola y2 = 4ax and line x = a in first quadrant. Curve y2 = 4ax and line x = a is plotted in adjacent figure. Enquiry: What about change in sign of area according to the position of curve. (Above or below x axis) Page 89

12 Area The area obtained by integration is positive. If the curve is above x-axis and b > a as in this figure. Area becomes negative if b > a and curve is below the x-axis. If for any value [a, b], the curve crosses the x-axis then the value of the integral gives the difference of areas of the portion of the curves lying below the x-axis and above the x-axis. As in the figure We can also write line the will give area with ve sign. We consider only numerical value. Find the area bounded by curve y = x(x 1)(x 2) and the x-axis. Here the curve is not having any standard shape. So we make a rough sketch. Solve with x-axis. Let s check where y is +ve or ve. When 0 < x < 1, y = +ve 1 < x < 2, y = ve Rough graph would be like this Shaded portion is the required area C O A 1 B 2 D Page 90

13 Area. Area bounded between the curves y = f(x) and y = g(x) and ordinates x = x1 and x = x2. Method: 1. To determine the area bounded between two curves. First find out points of intersection of the two curves. 2. If in the domain common to both (i.e. the domain given by the points of intersection) the curve lies above x axis, then area is Shaded portion (P1P2Q2Q1): y y P1 P2 f(x) y = f(x) g(x) y = g(x) O Q1 x = x1 Q2 x = x2 x x = x1 O x = x2 x If one part of graph or both the curves lie below x axis, then the individual integral must be g valuated according to previous knowledge. A = OMBN + OPCD Area enclosed between a curve y = f(x) and y-axis. y = mx B f(x) M N b O D a A P C 1 step: y = f(x) must be inverted to x = g(x) where g(x) = δ1(x) and Page 91

14 Area integral to be evaluated is y2 x = g(y) y1 Find area bounded by curve 2y = 2x x2 and x-axis. This is equation of parabola having vertex at (1, 0). First solve the with x-axis, i.e. solve and curve cuts the x-axis at x = 0 and x = 2 Required area = Area of portion OAB y B (1, 0) (2, 0) A O (0, 0). x Find the area of the region included between the parabola and the line Given parabola y And the given line is Q Solving (i) & (ii), (4, 12) ( 2, 3) P x Point of intersection are So, required are Find the area of the region included between the parabolas The equation of the given curve are Parabola and Solving (i) and (ii), Putting, where a > 0. y 2 x = 4ay y2=4ax (4a, 4a) from (ii) into (i) O (0, 0) Page 92 P x

15 Area. So, required area Find the area of the smaller region bounded by the ellipse and the straight line. The equation of the given curves are (0, b) (a, 0) is the equation of a straight line cutting x and y axes at (a, 0) and (0, b). Smaller region is bounded by two curve is shaded. Reg. area Page 93

16 Chapter 4 Differential Equation Differential Equation Day 1 An equation involving derivatives or differentials of one (or more) independent variables with respect to one (or more) independent variable(s) is called a differential equation i.e. it will be an equation is x, y and derivatives of y with respect to x. Order and degree of a differential equation: 1. The order of highest derivative involved in a differential equation is called order of differential equation. 2. The integer lower raised to highest derivative of a function is called the degree of a differential equation. order = 1, degree = 1 order = 4, degree = 1 equation should be free from related sign order = 2, degree = 1 Page 102

17 Differential Equation Self Efforts Find the degree and order of differential equation Solution 1. 4, , , not diff. Page 103

18 Differential Equation Linear & Non-Linear Differential equation: Equation is to be non-linear if 1. It s degree is more than one. 2. Any of the differential coefficient has exponent more than one. 3. Exponent of the dependent variable is more than one. 4. Products containing dependent variable and its differential coefficients are present. The differential equation is a non-linear differential equation, because its degree is 3, more than one. Page 104

19 Differential Equation Self Efforts Check whether linear or non-linear Solution 1. Linear 2. Non-linear 3. Linear Page 105

20 Differential Equation Solution of a differential equation: The solution of a differential equation is a relation between the variable involved which satisfies the differential equation. Show that is a solution of the differential equation We have Differentiating both sides w.r.t x, we get Differentiating w.r.t x, we get Thus, the function Hence, satisfies the differential equation is a solution of the given differential equation. Show that is a solution of the differential equation We have Differentiating w.r.t x, we get =y This shows that [Using (i)] is a solution of the given differential equation. Page 106