Monday, April 1, 2019

Definite integral

Definite inviolateDEFINITE underlyingIntegration is an grand concept in mathematics which, to get goingher with differentiation, forms one of the main operations in calculus. Given a cash in ones chips of a real covariant x and an interval a, b of the real name, the defined total, is defined informally to be the net sign ara of the region in the xy-plane bounded by the interpret of , the x-axis, and the vertical lines x = a and x = b.APPLICATIONS OF DEFINITE INTEGRALDefinite intacts argonnt just for area any more Any definite integral may be interpreted as a signed area. Area, volume, arc length, work, mass, fluid pressure, and compile financial value are quantities that may be calculated with definite integrals. The most important components of these problems are constructing the even out integral andInterpreting the results.nTWO VIEWS OF DEFINITE INTECRALWhen using the definite integral to solve various problems, it is useful to consider two different interpretations A limit of approximating sums The definite integral is formally defined as a limit of approximating sums using right sums. Accumulated change in an antiderivative The thorough Theorem of Calculus stateswhere F is any antiderivative of f on a b.The difference F(b) F(a) represents the accumulated change (or net change) in F over theinterval a b.To go the accumulated change in F over a b, integ valuate f, the rate function associated withF, over the interval a b.WHICH VIEW IS BETTER SUM OR ANTIDERIVATIVE ? Often we need to decide which view (or interpretation) of the definite integral is the amend one for a given application. It could be that an approximating sum is acceptable or that a precise typic antiderivative is more appropriate.If an integral is presented in symbolic form, then antidifferentiation seems commonsense. For data given graphically or in a table, approximating sums are the logical choice.EXPLANATIONTRAPEZOIDAL RULEThe trapezoidal overlook ( likewise known as the trapezoid precept, or the trapezium master in British English) is a way to approximately calculate the definite integralThe trapezoidal feel works by approximating the region under the graph of the function f(x) as a trapezoid and calculating its area. It follows thatTo calculate this integral more accurately, one commencement ceremony splits the interval of integration a,b into n littler subintervals, and then applies the trapezoidal rule on to each one of them. One obtains the entangled trapezoidal ruleIllustration of the composite trapezoidal rule (with a non-uniform grid)This offer alternatively be written asWhere(one potful also use a non-uniform grid).The trapezoidal rule is one of a family of formulas for numeric integration called NewtonCotes formulas. Simpsons rule is another, often more accurate, member of the very(prenominal) family. Simpsons rule and other like methods can be expected to break on the trapezoidal rule for functions which are twice continu ously differentiable however for rougher functions the trapezoidal rule is likely to prove preferable. Moreover, the trapezoidal rule tends to become extremely accurate when periodic functions are integrated over their periods, a fact best understood in connection with the EulerMaclaurin accession formula. For non-periodic functions, however, methods with unequally spaced points such as Gaussian quadrature and ClenshawCurtis quadrature are chiefly far more accurate ClenshawCurtis quadrature can be viewed as a change of variables to express arbitrary integrals in terms of periodic integrals, at which point the trapezoidal rule can be applied accuratelySIMPSON RULEIn numerical analysis, Simpsons rule is a method for numerical integration, the numerical approximation of definite integrals. Specifically, it is the following approximationSimpsons rule can be derived by approximating the integrand f(x) (in blue) by the quadratic interpolant P(x) (in red).METHODS BASED ON undetermined COEFFICIENTSNEWTON-COTES METHODS TRAPEZOIDAL METHODWe have n=1 , x0 =a , x1=b and h=x1-x0. Rn= (1)Using eq 1 ,the rule can be made exact for polynomial of score upto one.For f(x)=1 and x, we get the system of equations . f(x)= 1 x1- x0 = + or = + f(x) = x ( ) = + ( ) ( ) = + h( 2 + h ) = + ( ) h( 2 + h ) = ( + ) + h = h + h h= , or = From the first equation , we get h = h /2 . The method becomes = f( ) + f (The error constant is given by C = = 2 ( + 3 h + 3 + ) -2 -3 h -3h( + 2h + ) = SIMPSON S METHODWe have n = 2 , = a , = + h , = + 2h = b , h=(b a )/2 .We write = f( ) + f() + f( )The rule can be made exact for polynomials of degree upto two .For f(x) = 1, x , , we get the following system of equations. f(x) = 1 = + + , or 2h = + + (2)f(x) = x ( ) = + + -(3)f(x) = ( ) = + + (4)From (3) , we get ( ) ( ) = + + h) + + 2h) (2h) (2+ 2h) = ( + + ) + ( + 2 ) h = 2h + ( + 2 ) h 2h = + 2 (5)From (4) , we get ( + 6 h + 12 + 8 ) = + ( + 2 h + ) + ( + 4 + 2 h + ) h + ) Or h = + 4 (6)Solving (5) , (6) and (2) , we obtain = , = , The Method is given by .., = f() + 4 f() + f () The error constant is given byC = = COMPARISON between TRAPEZOIDAL RULE AND SIMPSONS RULETwo widely used rules for approximating areas are the trapezoidal rule and Simpsons rule. To motivate the new methods, we recall that rectangular rules approximated the function by a horizontal line in each interval. It is reasonable to expect that if we approximate the function more accurately inside each interval then a more efficient numerical proposal will follow. This is the idea behind the trapezoidal and Simpsons rules. Here the trapezoidal rule approximates the function by a suitably elect (not necessarily horizontal) line segment. The function values at the two points in the interval are used in the approximation. While Simpsons rule uses a suitably chosen parabolic shape (see Section 4.6 of the text) and uses the function at three points. The Maple scholarly person packa ge has look acrosss trapezoid and simpson that implement these methods. The command syntax is very equal to the rectangular approximations. See the examples below. Note that an even number of subintervals is required for the simpson command and that the default number of subintervals is n=4 for both trapezoid and simpson. with(student) trapezoid(x2,x=0..4) evalf(trapezoid(x2,x=0..4)) 22 evalf(trapezoid(x2,x=0..4,10)) 21.44000000 simpson(x2,x=0..4) evalf(simpson(x2,x=0..4)) 21.33333333 evalf(simpson(x2,x=0..4,10))21.33333333EXAMPLES OF TRAPEZOIDAL AND SIMPSONS RULEQuesEvaluate using trapezoidal and Simpsons Rule with h=0.05Sol x0= 1 , x1= 1.05 , x2= 1.1 , x3= 1.15 , x4= 1.20 , x5=1.25 , x6= 1.3I(trapezoidal) = = .05/2 f(1) + 2( f (1.05) + f(1.1) +f(1.15)+ f(1.120) +f (1.25)) +f(1.3) = 0.326808 = = =I(simpson) = f(1) + 4 (f (1.05)+ f(1.15) + f(1.25) + 2(f(1.1) + f(1.20) +f(1.3) = 0.321485Ques 2 Find the approximate value of I= Using (i) trapezoidal rule and ,(ii) Simpsons rule.Obtain a bound for the error. The exact value of I=ln2=0.693147 correct to six decimal places.Sol Using the Trapezoidal rule , I= ( 1+ ) = 0.75 misunderstanding = 0.75 0.693147 = 0.056853Using the Simpsons Rule, I= (1+ + ) = = 0.694444 Error = 0.694444 0.693147 = 0.001297

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