Definite integral

Definite intrinsical expressed INTEGRALIntegration is an important concept in mathematics which, in concert with differentiation, forms i of the main operations in calculus. Given a function of a real variable x and an interval a, b of the real line of products, the decided integral, is specify informally to be the net gestural area of the region in the xy-plane bounded by the graph of , the x-axis, and the vertical lines x = a and x = b.APPLICATIONS OF DEFINITE INTEGRALDefinite integrals arent just for area any more Any definite integral may be interpreted as a signed area. Area, volume, arc length, work, mass, fluid pressure, and accumulated financial cling to are quantities that may be calculated with definite integrals. The most important components of these problems are constructing the correct integral andInterpreting the results.nTWO VIEWS OF DEFINITE INTECRALWhen using the definite integral to solve various problems, it is useful to consider both different interpreta tions A limit of approximating sums The definite integral is formally defined as a limit of approximating sums using right sums. Accumulated reassign in an antiderivative The Fundamental 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 find the accumulated change in F over a b, integrate 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 correct one for a given application. It could be that an approximating sum is acceptable or that a precise symbolic antiderivative is more appropriate.If an integral is presented in symbolic form, then antidifferentiation seems reasonable. For data given graphically or in a table, approximating sums are the logical choice.EXPLANATIONTRAPEZOIDAL preceptThe trapezoidal line up (also k nown as the trapezoid regain, or the trapezium rule in British English) is a way to approximately calculate the definite integralThe trapezoidal rule 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 first splits the interval of integration a,b into n smaller subintervals, and then applies the trapezoidal rule on each of them. One obtains the composite trapezoidal rule case of the composite trapezoidal rule (with a non-uniform grid)This can alternatively be written asWhere(one can also use a non-uniform grid).The trapezoidal rule is one of a family of formulas for quantitative integration called NewtonCotes formulas. Simpsons rule is another, often more accurate, member of the same family. Simpsons rule and other like orders can be expected to improve on the trapezoidal rule for functions which are twice continuously differentiable however for rougher funct ions 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 substance formula. For non-periodic functions, however, methods with unequally spaced points such as Gaussian quadrature and ClenshawCurtis quadrature are generally 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 ME THODWe have n=1 , x0 =a , x1=b and h=x1-x0. Rn= (1) employ eq 1 ,the rule can be made exact for polynomial of degree 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 order is given by .., = f() + 4 f() + f () The error constant is given byC = = COMPARISON BETWEEN TRAPEZOIDAL RULE AND SIMPSONS RULE dickens 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 scheme will follow. This is the idea behind the trapezoidal and Simpsons rules. Here the trapezoidal rule approximates the function by a suitably chosen (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 parabolical shape (see Section 4.6 of the text) and uses the function at three points. The Maple student package has commands trapezoid and simpson that implem ent these methods. The command syntax is very similar to the rectangular approximations. hold in 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 val ue of I=ln2=0.693147 correct to six decimal places.Sol Using the Trapezoidal rule , I= ( 1+ ) = 0.75 Error = 0.75 0.693147 = 0.056853Using the Simpsons Rule, I= (1+ + ) = = 0.694444 Error = 0.694444 0.693147 = 0.001297