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MATHEMATICS FORM FIVE TOPIC 10: INTEGRATION

 

 Integration :Is the reverse process of differentiation, i.e. the process of finding the expression for y in terms of x when given the gradient function.

The symbol for integration is

This is the general power of integration it works for all values of n except for n = -1

Example

1.     

2.       Integrate the following with respect to x
(i)3x2

Solution

  Integration of constant

Integration by change of variables

If x is replaced by a linear function of x, say of the form ax + b, integration by change of variables will be applied
E.g

Example

Note: 2x  is the derivative of x2 + 1 in this case substitution is useful

i.e. let u = x2 + 1


This converts into the form      

Standard integrals

· 

EXERCISE

Find the integral of the following functions

gration by partial fraction

Integration by partial fraction is applied only for proper fraction

E.g. 

Note that:

Improper fraction

If the degree of numerator is equal or greater than of denominator, adjustment must be made

Example

1.      Find 

Solution


If the denominator doesn’t factorize, splitting the numerator will work

→ Numerator = A (derivative of denominator) + B

Example

Solution

Important

It can be shown that

EXERCISE

     

Integrated of the form


Note that:

1. If the denominator has two real roots use partial fraction

2. If the denominator has one repeated root use change of variable or recognition

3. If the denominator has no real roots, use completing the square


Integration of Trigonometric Expression
Integration of Even power of

Note that

For integrand containing   and  , or even powers of these, the change of variable    can be used.


APPLICATION OF INTEGRATION

To determine the area under the curve


= f (b) – f (a)

Examples

1. Find the area under the curve f(x) =x2+1 from x=0 to x=2

2. Find the area under the curve f(x) = from x=1 to x=2

3. Find the area bounded by the function f(x) =x 2-3, x=0, x=5 and the x- axis

Solution

  1. f(x) =  + 1

y intercept=1

EXERCISE

1.      Find the area between y = 7-xand the x- axis from x= -1 to x=2

2.      Find the area between the graph of y=xx – 2 and the x- axis from x= -2 to x=3

Solution

1.        y =7-x2

Where y- intercept =7
= 6.67 + 11.3

=17.97sq units

Volume of the Solids of Revolution

The volume,V of the solid of revolution is obtained by revolving the shaded portion under the curve, y= f(x) from x= a to x =b about the x -axis is given by

Example 1

Find the volume of revolution by the curve y=x2 from x=0 to x=2 given that the rotation is done about the the x- axis

Exercise

1. Find the volume obtained when each of the regions is rotated about the x – axis.

a) Under  y= x3, from x =0 to x=1
b) Under y2= 4-x, from x=0 to x=2
c)Under y= x2, from x=1 to x=2
d)Under y= √x, from x=1 to x=4

2. Find the volume obtained when each of the region is rotated about the y-axis.
a) Under y= x2, and the y-axis from x=0 to x=2
b) Under y= x3, and the y-axis from y=1 to y=8
c) Under y= √x, and the y-axis from y=1 to y=2

 LENGTH OF A CURVE

Consider the curve

Example

Find the length of the part of the curves given between the limits:

 

Mathematics (from Ancient Greek μάθημα; máthēma: ‘knowledge, study, learning’) is an area of knowledge that includes such topics as numbers (arithmetic, number theory), formulas and related structures (algebra), shapes and the spaces in which they are contained (geometry), and quantities and their changes (calculus and analysis).

Most mathematical activity involves the use of pure reason to discover or prove the properties of abstract objects, which consist of either abstractions from nature or—in modern mathematics—entities that are stipulated with certain properties, called axioms. A mathematical proof consists of a succession of applications of some deductive rules to already known results, including previously proved theorems, axioms and (in case of abstraction from nature) some basic properties that are considered as true starting points of the theory under consideration.

Mathematics is used in science for modeling phenomena, which then allows predictions to be made from experimental laws. The independence of mathematical truth from any experimentation implies that the accuracy of such predictions depends only on the adequacy of the model. Inaccurate predictions, rather than being caused by incorrect mathematics, imply the need to change the mathematical model used. For example, the perihelion precession of Mercury could only be explained after the emergence of Einstein’s general relativity, which replaced Newton’s law of gravitation as a better mathematical model.

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