Day 1 - Online · 2014-11-12 · Kaysons Education Indefinite Integral Page 3 15 16 17 What are the...

Kaysons Education 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 is called integration

If

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

Indefinite Integral Chapter

1


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:

1

2

3

4

5

6

7

8

9

10 –

11

12

13

14


What are the various Operations on Integration

1.

2.

Illustration

Illustration

15

16

17

18

19

20


Self Efforts

Solution


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.

Illustration

Illustration

Illustration

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 .

Kaysons Education Definite Integral

8

Day – 1

What is Definite Integral?

Let be a function of x defined in the closed interval [a, b] and be another function, such that

for all x in the domain of then

Where 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.

Illustration

Definite Integral Chapter

2


Page 69

Self Efforts

Solution


Page 70

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.

Illustration

Illustration

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 71

(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]

Illustration

—

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 72

Illustration

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

Illustration

Kaysons Education Area

Page 89

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

Illustration

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)

Definite Integral Chapter

3


Page 90

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.

Illustration

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

B

2

D

A

C

O 1


Page 91

.

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):

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.

1 step: y = f(x) must be inverted to x = g(x) where g(x) = δ1(x) and

P1 P2

Q2 Q1

y = g(x)

y = f(x)

x

y

O x = x1 x = x2

f(x)

x

y

O

g(x)

x = x1 x = x2

a O b

D P

C

B

A

N

M

f(x)

y = mx


Page 92

integral to be evaluated is

Illustration

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

.

Illustration

Find the area of the region included between the parabola and the line

Given parabola

And the given line is

Solving (i) & (ii),

Point of intersection are

So, required are

Illustration

Find the area of the region included between the parabolas , where a > 0.

The equation of the given curve are

Parabola

and

Solving (i) and (ii), Putting from (ii) into (i)

x = g(y)

y1

y2

(2, 0)

(1, 0)

(0, 0) O A

B

y

x

(2, 3)

y

x

(4, 12) P

Q

y

x P O

(0, 0)

(4a, 4a)

y2=4ax

x2 = 4ay


Page 93

.

So, required area

Illustration

Find the area of the smaller region bounded by the ellipse and the straight line

.

The equation of the given curves are

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

Illustration

(a, 0)

(0, b)

Kaysons Education Differential Equation

Page 102

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.

Illustration

order = 1, degree = 1

Illustration


Illustration

equation should be free from related sign


Differential Equation Chapter

4


Page 103

Self Efforts

Find the degree and order of differential equation.

1. 2.

3.

Solution

1. 4, 2 2. 2, 3 3. 2, not diff.


Page 104

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.

Illustration

The differential equation is a non-linear differential equation, because its

degree is 3, more than one.


Page 105

Self Efforts

Check whether linear or non-linear

1. 2.

3.

Solution

1. Linear 2. Non-linear 3. Linear


Page 106

Solution of a differential equation:

The solution of a differential equation is a relation between the variable involved which satisfies

the differential equation.

Illustration

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 satisfies the differential equation

Hence, is a solution of the given differential equation.

Illustration

Show that is a solution of the differential equation

We have

Differentiating w.r.t x, we get

= y [Using (i)]

This shows that is a solution of the given differential equation.

Day 1 - Online · 2014-11-12 · Kaysons Education Indefinite Integral Page 3 15 16 17 What are the...

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