Jingle bell

2. PRESENTED BY ,
GOWTHAM.S - 15BME110

3. INTEGRAL CALCULUS :
It is the branch of calculus which deals
with functions to be integrated.
INTEGRATION :
 Integration is the reverse process of
differentiation.
 The function to be integrated is referred to
as integrand while the result of an integration
is called integral.
 The integral is equivalent to the area under
the curve.

4. The integral symbol is an elongated S –
denoting sum, was introduced by Leibniz, who
named integral calculus as calculus
summatorious.
Numerical integration is carried by the
numerical methods and they are of three types:
Trapezoidal rule
Simpson’s 1 st rule
Simpson’s 2 nd rule

5. DEFINITE INTEGRAL : defined by the limit
values a & b of the independent variable.
INDEFINITE/PRIMITIVE INTEGRAL :
An integral with no restrictions imposed
on its independent variable.

6.  It is applicable for equal intervals.
 The error is of order h2.
 The accuracy can be improved by increasing
the no. of intervals & by decreasing the value
of h.
 In this rule, y(x) is a linear function of x.
 In general, trapezoidal rule is less accurate
when compared with Simpson's rule.

7.  It is also known as Simpson's one-third (1/3)
rule.
 It is applicable for even intervals.
 The error is of order h4.
 In this rule, y(x) is a polynomial of degree 2.
 It uses 3 data points.

8.  It is also known as Simpson's 3/8 th rule.
 The error is of order h5.
 In this rule, y(x) is a polynomial of degree 3.
 It is applicable for the intervals which is
multiple of 3.
 It uses four data points.

9. TRAPEZOIDAL RULE :
SIMPSON’S 1 st RULE :
𝑥0
𝑥𝑛
𝑓( 𝑥) 𝑑𝑥 =
ℎ
3
[ ( y0 + yn ) + 2( y2 + y4 +…) +
4(y1 + y3+…) ]
SIMPSON’S 2 nd RULE :
𝑥0
𝑥𝑛
𝑓 (𝑥) 𝑑𝑥 =
3ℎ
8
[ 𝑦0 + 𝑦𝑛 + 3 𝑦1 + 𝑦2 + 𝑦4 + 𝑦5 + ⋯
+ 2 𝑦3 + 𝑦6 + 𝑦9 + ⋯ ]
𝑥0
𝑥𝑛
𝑓 𝑥 𝑑𝑥 =
ℎ
2
[ 𝑦0 + 𝑦𝑛 + 2(𝑦1 + 𝑦2 + ⋯ + 𝑦𝑛 − 1)]

10. Where…
x0 = initial value of x,
y0 = initial value of y,
xn = final value of x,
yn = final value of y,
h = interval distance,
h =
𝑏−𝑎
𝑛
n = no. of intervals.

11. It helps to
 Find the area.
 Locate the centroid.
 Find the arc length of a graph.
 Find the surface area of a solid.
 Find the volume of a solid figure.
 Solve for the work done.
 Solve the moment of inertia.

12. It is also used to find
 Sectional area.
 Waterplane area.
 Submerged volume.
 Longitudinal center of floatation (LCF).
 Vertical center of buoyancy (VCB).

13. A Small Waterplane Area Twin Hull, better
known by the acronym SWATH, is a twin-hull
ship design that minimizes hull cross
section area at the sea's surface.
Minimizing the ship's volume near the
surface area of the sea, where wave energy is
located, maximizes a vessel's stability, even in
high seas and at high speeds.

14.  


area
Lpp
WP dx
x
y
dA
A
0
)
(
2
2
)
width(
al
differenti
)
(
at
breadth)
-
f
offset(hal
)
(
)
area(
al
differenti
)
area(
plane
water
2
2
ft
dx
ft
x
y
x
y
ft
dA
ft
AWP





15.  In geology, the structure of the interior of
a planet is often illustrated using a diagram
of a cross section of the planet that passes
through the planet's centre, as in the cross
section of Earth.
 Cross sections are often used in anatomy to
illustrate the inner structure of an organ, as
shown at left.
 A cross section of a tree trunk, reveals growth
rings that can be used to find the age of the
tree and the temporal properties of its
environment.

16. z
y
dz
y(z)
T
 


area
T
t dz
z
y
dA
A
0
sec )
(
2
2
)
width(
al
differenti
)
z(
at
breadth)
-
f
offset(hal
)
(
)
area(
al
differenti
)
(
to
up
area
sectional
2
2
sec
ft
dz
ft
y
z
y
ft
dA
ft
z
A t





17. An object that sinks displaces an amount of
fluid equal to the object's volume. Thus buoyancy is
expressed through Archimedes' principle, which
states that the weight of the object is reduced by its
volume multiplied by the density of the fluid. If the
weight of the object is less than this displaced
quantity, the object floats; if more, it sinks. The
amount of fluid displaced is directly related (via
Archimedes' Principle) to its volume.
In the case of an object that sinks (is totally
submerged), the volume of the object is displaced.
In the case of an object that floats, the amount of
fluid displaced will be equal in weight to the
displacing object .

18. SUBMERGED VOLUME : Integration of sectional
area over the length of ship.
x
y
)
(x
As
 
n
n
s y
y
y
y
x 






 1
2
1
0 4
..
2
4
y
3
1

19. It is used to find the center of water plane
area (i.e.) distance from reference point to
center of floatation.
+
+
-
FP
WL

20. 1. Start with a picture of what you are about to integrate.
2. Show the differential element you are using.
3. Properly label your axis and drawing.
4. Write out the generalized calculus equation written in
the same symbols you used to label your picture .
5. Write out Simpson’s equation in generalized form.
6. Substitute each number into the generalized
Simpson’s equation.
7. Calculate final answer.

21. Numerical methods is not
an elegant subject. It is a
lot of technical detail and
dirty work. But it is the way
we solve most of the real
world problems.