MATHS TERM PAPER ON What are the applications of definite integral.compare 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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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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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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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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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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= 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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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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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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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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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+b) / 2. One can use Lagrange polynomial interpolation to find an expression for this polynomial,

An easy (albeit tedious) calculation shows that

This calculation can be carried out more easily if one first observes that (by scaling) there is no loss of generality in assuming that a

= 1 and b = 1.

Averaging the midpoint and the trapezoidal rules:-

Another derivation constructs Simpson's rule from two simpler approximations: the midpoint rule

and the trapezoidal rule

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The errors in these approximations are

respectively. It follows that the leading error term vanishes if we take the weighted average

This weighted average is exactly Simpson's rule.

Using another approximation (for example, the trapezoidal rule with twice as many points), it is possible to take a suitable

weighted average and eliminate another error term. This is Romberg's method.

Undetermined coefficients:-

The third derivation starts from the ansatz

The coefficients , and can be fixed by requiring that this approximation be exact for all quadratic polynomials. This yields

Simpson's rule.

Error:-

The error in approximating an integral by Simpson's rule is

where is some number between a and b

The error is asymptotically proportional to (b a)5. However, the above derivations suggest an error proportional to (b a)4.Simpson's rule gains an extra order because the points at which the integrand is evaluated are distributed symmetrically in the

interval [a, b].

Note that Simpson's rule provides exact results for any polynomial of degree three or less, since the error term involves the fourth

derivative of f.

Composite Simpson's rule:-

If the interval of integration [a,b] is in some sense "small", then Simpson's rule will provide an adequate approximation to the exact

integral. By small, what we really mean is that the function being integrated is relatively smooth over the interval [a,b]. For such a

function, a smooth quadratic interpolant like the one used in Simpson's rule will give good results.

However, it is often the case that the function we are trying to integrate is not smooth over the interval. Typically, this means that

either the function is highly oscillatory, or it lacks derivatives at certain points. In these cases, Simpson's rule may give very poor

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results. One common way of handling this problem is by breaking up the interval [a,b] into a number of small subintervals.

Simpson's rule is then applied to each subinterval, with the results being summed to produce an approximation for the integral over

the entire interval. This sort of approach is termed the composite Simpson's rule.

Suppose that the interval [a,b] is split up in n subintervals, with n an even number. Then, the composite Simpson's rule is given by

where xj = a + jh for j = 0,1,...,n 1,n with h = (b a) / n; in particular, x0 = a and xn = b. The above formula can also bewritten

as

The error committed by the composite Simpson's rule is bounded (in absolute value) by

where h is the "step length", given by h = (b a) / n.

This formulation splits the interval [a,b] in subintervals of equal length. In practice, it is often advantageous to use subintervals of

different lengths, and concentrate the efforts on the places where the integrand is less well-behaved. This leads to the adaptive

Simpson's method.

Alternative extended Simpson's rule:-

This is another formulation of a composite Simpson's rule: instead of applying Simpson's rule to disjoint segments of the integral to

be approximated, Simpson's rule is applied to overlapping segments.

Simpson's 3/8 rule:-

Simpson's 3/8 rule is another method for numerical integration proposed by Thomas Simpson. It is based upon a cubic interpolation

rather than a quadratic interpolation. Simpson's 3/8 rule is as follows:

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The error of this method is:

where is some number between a and b. Thus, the 3/8 rule is about twice as accurate as the standard method, but it uses one more

function value. A composite 3/8 rule also exists, similarly as above.

References:-

1)http://en.wikipedia.org/wiki/Trapezoidal_rule

2)http://en.wikipedia.org/wiki/Simpson's_rule

3)http://www.mecca.org/~halfacre/MATH/appint.htm

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http://en.wikipedia.org/wiki/Trapezoidal_rulehttp://en.wikipedia.org/wiki/Simpson's_rulehttp://www.mecca.org/~halfacre/MATH/appint.htmhttp://en.wikipedia.org/wiki/Trapezoidal_rulehttp://en.wikipedia.org/wiki/Simpson's_rulehttp://www.mecca.org/~halfacre/MATH/appint.htm

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

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