MATHS TERM PAPER ON What are the applications of definite trapezoidal rule and simpson rule

8/8/2019 MATHS TERM PAPER ON What are the applications of definite integral.compare trapezoidal rule and simpson rule.

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Applications of Definite Integral

Topic:- What are the applications of definite integral.compare trapezoidal rule and simpson rule.

Submitted to:-Mr Rohit Gandhi

Subject code:-MTH-204

Subject:-Numerical analysis

Submitted by:-Chandan dhir

Roll no:-RC2802A05

Sec no:-RC2802

Reg no:-10801530

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Applications of Definite Integral

Areas 2

Arc Length 3

Volumes of Solids of Revolution 4

Area of Surface of Revolution 5

Definiteintegral 6

Trapezoidal rule 8

Composite trapezoidal rule 10

Simpsons rule 11

Composite simpsons rule 12

3/8 simpsons rule 13

Areas

A Equations of Curves are represented in Rectangular Form

Let A denote the area ( or total area) of the shaded region.

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Applications of Definite Integral

Theorem The area enclosed by the graph of )x(fy = , the x-axis and the lines ax = and bx = is equal todxba )x(f or dxba y .

Theorem The area enclosed by the graph of )y(gx = , the x-axis and the lines cy = and dy = is equal todydc )y(g or dydc x .

B Equations of Curves are in parametric Form

It is known that the area between the curve )x(fy = and the lines ax = , bx = and 0y = is given bydxba y .

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Applications of Definite Integral

If the equation of the curve is in parametric form ==

)t(Gy

)t(Fx,

where t is a parameter, and if,bxwhent

;axwhent

==

)t('Fdt

dx = is a continuous function on ],[ , and )t('F does not change sign is in ),( , then thearea of the region bounded by the curve =

=)t(Gy

)t(Fx, the x-axis and the lines ax = , bx = is

dtdt )t('xy)t('F)t(G

= . ( Integration by substitution )This formula is also true when if > . In this case 0)t('F

dt

dx for all ),(t .

Arc Lengths

A Equations of curves are in Rectangular Form

Theorem If a curve )x(fy = has a continuous derivative on ]b,a[ , then the length of the curve from ax = tobx = is given by dx

dx

dydx +

b

a

2b

a

21))x('f(1 .

Remark If the equation of the curve is in the form )y(gx = , then the length of the arc between cy = and dy = isgiven by dy

dy

dxdy +dc 2dc 2 1))x('g(1 .

B Equations of Curves are in Parametric Form

Theorem When a function is expressed in parametric form )t(fx = and )t(gy = , the arc length s of the curvefrom at = to bt = is given by

dt +ba 22 ))t('g())t('f(s

Volumes of Solids of Revolution

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Applications of Definite Integral

A Disc Method

Theorem Let )x(fy = be a continuous function defined on ]b,a[ , and S be the region bounded by the curve)x(fy = , the lines ax = , bx = and the x-axis. Then the volume V of the solid generated by revolving the

region S one complete revolution about the x-axis is given by

[ ] dxdx 2ba

b

a

2)x(fyV = .

Remark In parametric form

==

)t(gy

)t(hx, the volume of solid of resolution generated by revolving the region enclosed

by the graph, x-axis and from 1t to 2t about x-axis.

dt)t('h))t(g(V2

1

t

t

2Theorem Let )x(fy = be a continuous function defined on ]b,a[ , and S be the region bounded by the curve

)x(fy = , the lines ax = , bx = and hy = . Then the volume V of the solid generated by revolving theregion S one complete revolution about the hy = is given by

[ ] dxdx 2ba

b

a

2h)x(f)hy(V

B Shell Method

Theorem Let f be a function continuous on ]b,a[ . If the area bounded by the graph of )x(f , the x-axis and the

lines ax = and bx = is revolved about the y-axis, the volume V of solid generated is

b

a)x(xf2 dx

Area of Surface of Revolution

Theorem Suppose )x(fy = has a continuous derivative on ]b,a[ . Then the area S of the surface ofrevolution by the arc of the curve )x(fy = between ax = and bx = about the x-axis is

S = [ ] dx2ba

)x('f1)x(f2 += dx

2b

a dx

dy1y2 +

Remark The corresponding formula for the area of the surface of revolution obtained by revolving an arc of a curve

)y(gx = from cy = to dy = about the y-axis isS = [ ] dy2d

c)y('g1)y(g2 +

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Applications of Definite Integral

= dy

2d

c dy

dx1x2

+

Theorem If a portion of the curve of parametric equations )t(xx = , )t(yy = between the points corresponding to1t and 2t is revolved about the x-axis, the surface area S is

S = dtds +21

2

1

t

t

22t

t)]t('y[)]t('x[)t(y2)t(y2 .

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Applications of Definite Integral

Defifnite integral:-

A definite integral is an integral (1)

with upper and lower limits. If is restricted to lie on the real line, the definite integral is known as a Riemann integral

(which is the usual definition encountered in elementary textbooks). However, a general definite integral is taken in the

complex plane, resulting in the contour integral (2)

with , , and in general being complex numbers and the path of integration from to known as a contour.

The first fundamental theorem of calculus allows definite integrals to be computed in terms of indefinite integrals, sinceif is the indefinite integral for a continuous function , then (3)

This result, while taught early in elementary calculus courses, is actually a very deep result connecting the purely

algebraic indefinite integral and the purely analytic (or geometric) definite integral. Definite integrals may be evaluatedin Mathematica using Integrate[f, x, a, b].

The question of which definite integrals can be expressed in terms of elementary functions is not susceptible to any

established theory. In fact, the problem belongs to transcendence theory, which appears to be "infinitely hard." For

example, there are definite integrals that are equal to the Euler-Mascheroni constant . However, the problem of

deciding whether can be expressed in terms of the values at rational values of elementary functions involves the

decision as to whether is rational or algebraic, which is not known.

Integration rules of definite integration include (4)

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Applications of Definite Integral

and (5)

For , (6)

If is continuous on and is continuous

Trapezoidal Rule:-

This article is about the quadrature rule for approximating integrals. For the Explicit trapezoidal rule for solving initial value

problems, see Heun's method.The function f(x) (in blue) is approximated by a linear function (in red).In mathematics, the

trapezoidal rule (also known as the trapezoid rule or trapezium rule) is an approximate technique for calculating the definite

integral.The trapezoidal rule works by approximating the region under the graph of the function f(x) as a trapezoid and calculating

its area. It follows that

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Applications of Definite Integral

Composite trapezoidal rule:-

Composite trapezoidal rule

Illustration of the composite trapezoidal rule (with a uniform grid)

To calculate this integral more accurately, one first splits the interval of integration [a,b] into N smaller, uniform subintervals, and

then applies the trapezoidal rule on each of them. One obtains the composite trapezoidal rule:

This can alternatively be written as:

Where

This formula is not valid for a non-uniform grid, however the composite trapezoidal rule can be used with a variable trapezium

widths.

Non-uniform intervals:-Given data and then the integral can be approximated as follows,

where

yi = f(xi).

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Applications of Definite Integral

Simpson's rule:-In numerical analysis, Simpson's rule is a method for numerical integration, the numerical approximation of

definite integrals. Specifically, it is the following approximation:

The method is credited to the mathematician Thomas Simpson (17101761) of Leicestershire, England.

Derivation:-

Simpson's rule can be derived in various ways.

Quadratic interpolation:-

One derivation replaces the integrand f(x) by the quadratic polynomial P(x) which takes the same values as f(x) at the end points a

and b and the midpoint m = (a