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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+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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Applications of Definite Integral
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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Applications of Definite Integral
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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Applications of Definite Integral
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