numerical integration and its applications

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PRESENTED BY ,

GOWTHAM.S - 15BME110

NUMERICAL INTEGRATION AND ITS APPLICATIONS :

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.

NUMERICAL INTEGRATION :

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 ruleSimpson’s 1 st ruleSimpson’s 2 nd rule

NUMERICAL INTEGRTION :

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.

NUMERICAL INTEGRATION :

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.

TRAPEZOIDAL RULE :

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.

SIMPSON’S 1 st RULE :

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.

SIMPSON’S 2 nd RULE :

TRAPEZOIDAL RULE :

SIMPSON’S 1 st RULE :

= [ ( y0 + yn ) + 2( y2 + y4 +…) + 4(y1 + y3+…) ]

SIMPSON’S 2 nd RULE :

=

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 = n = no. of intervals.

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.

APPLICATION OF NUMERICAL INTEGRATION :

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

APPLICATION of NUMERICAL INTEGRATION :

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.

WATERPLANE AREA :

WATERPLANE AREA :

area

LppWP dxxydAA

0 )( 2 2

) width(aldifferenti )(at breadth)-foffset(hal )(

)area( aldifferenti

)area( planewater 2

2

ftdxftxyxy

ftdA

ftAWP

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.

APPLICATION OF SECTIONAL AREA :

SECTIONAL AREA :

z

y

dz

y(z)T

area

T

t dzzydAA0sec )( 2 2

) width(aldifferenti )z(at breadth)-foffset(hal )(

)area( aldifferenti

)( toup area sectional2

2sec

ftdzftyzy

ftdA

ftzA t

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 .

SUBMERGED VOLUME :

SUBMERGED VOLUME : Integration of sectional area over the length of ship.

SUBMERGED VOLUME : (LONGITUDINAL INTEGRATION)

x

y)(xAs

nns yyyyx 1210 4..24y 31

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

LONGITUDINAL CENTRE OF FLOATATION (LCF) :

+

+-FP

WL

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.

CALCULATIONS FOR SIMPSON’S EQUATION :

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.

THANK YOU