Integration

Engineering Mathematics-

Integration

LEARNING OBJECTIVES

• To understand the virtual loss of GM and the calculations.

• To calculate the maximum trim allowed to maintain a minimum stated GM.

• To understand the safe requirements for a ship prior enter into dry dock. To understand the critical period during dry docking process.

Course Outline

• Name of Course :Chief and Second

Engineer 3000 kW or more (Unlimited

Voyage)

• Course Code/Module : ECSU , Part

A

• Subject : Mathematics and

Engineering Drawing

Course Outline

Module Aims

• To provide students with the familiarization to the fundamentals of calculus Mathematics required for engineering practice and problem solving.

General Learning Objective - GLO

• Recognize that integration is the inverse process of differentiation, and apply this knowledge to determine the area/volume/work done.

Specific Learning Objectives - SLO

Recognize that integration can be considered the reverse of differentiation

process.

Explain the integration of x, trigo. functions, 1/x, exponential functions.

Evaluate the constant of integration.

Perform the definite Integral.

Apply integration to find:

• a. Area under curves.

• Volume of solid revolution

• Work done

• Mean & root mean square (rms) values

• Centroid

Course outline • Instructional Hours

• Lecture : 40 hours

• Topics Hours

• Integration as reverse of differentiation 2

• Integration of functions: x,Trig,1/x, Exponential 8

• Evaluation of constant of integration 4

• Definite integral 6

• Application of integral calculus to: 20

a. Area under curves.

b. Volume of solid revolution

c. Work done

d. Mean & root mean square (rms)

e. valuesCentroid20

Course Outline

Integration as the Process of Summation

Integration as the Reverse of Differentiation

Integration of functions

Applications of Integration : Areas Bounded

by Curves and Volumes of Revolution

• Teaching Methods -

• Combination of combination of methods as necessary -lectures, practice

• Assessment Methods

• Lecturer Class Assessment 1 20 %

• Lecturer Class Assessment 2 20 %

• Lecturer Class Assessment 3 20 %

• Final Exam 40 %

• Recommended Texts

• K A Stroud (1992), Engineering Mathematics Programmes And Problems

• G.S.Sharma & I.J.S.Sarna (1992), Engineering Mathematics

We know how to find the area of simple

geometric shapes such as the triangle below

Integration : Concept and Theory

y

x

21

1

2

But how do we find the are of geometric object

which do not have straight edges ?

ba

x

y

f(x)

Integration : Concept and Theory

So, how do we go about finding the area under

the curve f(x), between x=a and x=b ?

Well,

we can divide the area under the curve into

separate rectangles …

… find the area of each rectangle …

… and then sum these areas in order to find an

approximate answer to area under curve

Integration : Concept and Theory

Find area of each rectangle …

… then sum all areas between x=a and x=b

Integration : Concept and Theory

ba

x

y

f(x)

h

Process of Integration

• Integration is reverse of differentiation

• In differentiation, if f(x)= then f`(x)= 4x . Thus the integral of

• integration is the process of moving from f`(x) to f(x). By similar

reasoning, the integral of.

• Integration is a process of summation or adding parts together and an

elongated S, shown as, is used to replace the words ‘the integral of’. Hence,

from above,

‘c’ is called the arbitrary constant of integration

Integrationis the reverse process of differentiation.

Power series integration,

increase the exponent by one

- (PNO2)-n+1

and divide by the

new exponent.

-n+1

k t

1= + C

Constant

of integration

- (PNO2)-n dPNO2 = k dt

The Fundamental Theorem of Calculus

If then f x a

b

dx F b F a .F' x f x ,

If we know an anti-derivative, we can use it to find the

value of the definite integral.

The Fundamental Theorem of Calculus

If then f x a

b

dx F b F a .F' x f x ,

If we know an anti-derivative, we can use it to find the

value of the definite integral.

If we know the value of the definite integral, we can use it

to find the change in the value of the anti-derivative.

Integration of function

• The general solution of integrals of the form

• axndx, where a and n are constants is given by:

This rule is true when n is fractional, zero, or a

positive or negative integer, with the exception of n = -1.

).1( 1

1

nCn

xdxx

nn

Standard Integrals

• The integral of constant K is kx+c. for

example

Integral of several terms = Sum of integral of the separate terms

for exemple

Problems

Exercise