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

Apr 10, 2018

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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

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