Assignment, Business Mathematics II, BBA-BI 2nd semester, Ace Institute of Management

1. By:
•Anmol Pasa Shrestha
•Chhitiz Shrestha
•Leeza Shrestha
•Niraj Taujale
•Raj Shrestha
•Tenzing Tashi
•Zhang Peng

2. Introduction
 The relationship involving the rate
of change of two variables,
 but also needed to know the direct
relationship between the two
variables
 For example, we may know the
velocity of an object at a particular
time, but we may want to know the
position of the object at that time.
 To find this direct relationship, we
need to use the process which is
opposite to differentiation. This is
called integration (or anti-
differentiation).
 The processes of integration are
used in many applications

3. Background
 An important concept in
mathematics,
 Defined informally to be the net
signed area of the region in the xy-
plane bounded by the graph of
ƒ, the x-axis, and the vertical lines
x = a and x = b
 May also refer to the notion of
antiderivative, a function F whose
derivative is the given function ƒ
 the basic tools of calculus, with
numerous applications in
science, business and engineering

4. Background
 Example
 x = 0 to x = 1 and y = f(0) = 0 and
y = f(1) = 1.
 Its area is exactly 1.
 Decreasing the width of the
approximation rectangles shall
give a better result; so cross the
interval in five steps, using the
approximation points
0, 1⁄5, 2⁄5, and so on to 1.
 Thus √1⁄5, √2⁄5, and so on to
√1 = 1.

5. Background
 Basic knowledge of derivatives is a must
 Definition:
 The differential of y = f(x) is written: dy = f '(x)dx.
 Example:
 Find the differential of y = 3x5- x.
 Answer
 dy = f'(x)dx
 dy = (15x4 - 1)dx

6. History
 Integration can be traced as far back
as ancient Egypt before 1800 BC
 Further developed and employed by
Archimedes and used to calculate
areas for parabolas
 Similar methods were independently
developed in China around the 3rd
century AD by Liu Hui
 Next major step in integral calculus
came in Iraq when the 11th century
mathematician Ibn al-Haytham
(known as Alhazen in Europe
 Also formulated independently by
Isaac Newton and Gottfried Leibniz
in the late 17th century

7. History
 Acquired a firmer footing
with the development of
limits and was given a
suitable foundation by
Cauchy in the first half of
the 19th century,
 Integration was first
rigorously formalized, using
limits, by Riemann ,
 Other definitions of
integral, extending
Riemann's and Lebesgue's
approaches, were
proposed.

8. Techniques of integration
 Various techniques of
integration
 Integration by general
rule
 Integration by exponential
form
 Integration by parts
 Integration by substitution
 Etc.

9. Methods of integration
 The General Power Formula:
 Example:

10. Methods of integration
 The Basic Logarithmic Form:
 Example

11. Methods of integration
 The Exponential Form:
 Example

12. Methods of integration
 Integration by Parts:
 Example
 u = ln xdv = dxv = x

13. Methods of integration
 Integration by Substitution:
 Example

14. Application of Integration
 The Petronas Towers in Kuala Lumpur
experience high forces due to winds.
Integration was used to design the
building for strength.
 The Sydney Opera House is a very
unusual design based on slices out of a
ball. Many differential equations (one
type of integration) were solved in the
design of this building.
 Historically,one of the first uses of
integration was in finding the volumes of
wine-casks (which have a curved
surface).

15. Benefits of integration in
Business
 Introduce new applications
and technologies more
efficiently and at a lower cost
 More easily modify and
automate business processes
to meet new needs
 Provide more delivery
channels for your organization
 Replacing batch processing
with real-time communication
 Linking back-office systems to
new applications .
 Sharing data between System

16. Benefits of integration in
Business
 Commercial organizations use mathematics
in
 accounting,
 Inventory management,
 Marketing,
 Sales forecasting, and
 Financial analysis

17. PU Board Questions (2008)

18. PU Board Questions (2009)

19. Conclusion
 Very important mathematical tool
 Used in many fields
 Important in business
 Helps to estimate things like
 Marginal cost
 Marginal revenue
 Profit
 Gross loss
 Etc.