Chapter 4 Numerical Integration andNumerical DifferentiationSectionl the Introduction of Practical ProblemSection2 Mechanical Quadrature Method andAlgebraic PrecisionSection3 Newton-Cotes Quadrature FormulaSection4 Compound MultiplicativeSection5 Romberg Quadrature FormulaSection6 Gaussian Quadrature FormulaSection7 Numerical Differentiation上页下页返圆
上页 下页 返回 Chapter 4 Numerical Integration and Numerical Differentiation Section1 the Introduction of Practical Problem Section3 Newton—Cotes Quadrature Formula Section4 Compound Multiplicative Section2 Mechanical Quadrature Method and Algebraic Precision Section5 Romberg Quadrature Formula Section6 Gaussian Quadrature Formula Section7 Numerical Differentiation
s1 the Introduction of Practical ProblemShenzhou-VI spacecraft ran 5 laps on the elliptical orbit, whichorbit inclination angle is 42.4 degrees , height of perigee 200 kmheight of apogee 347km. Try to calculate the travel kilometers ofShenzhou-VI.the perimeter of elliptical orbit is the main factor上页下页返圆
上页 下页 返回 §1 the Introduction of Practical Problem Shenzhou-VI spacecraft ran 5 laps on the elliptical orbit, which orbit inclination angle is 42.4 degrees , height of perigee 200 km, height of apogee 347km. Try to calculate the travel kilometers of Shenzhou-VI. the perimeter of elliptical orbit is the main factor
According to the elliptical parametric equation and arclength formula of curve. we can calculate the perimeter ofelliptical orbit2Tcos? tdtL = 4a2aThis is a definite integral, we can just work out itsvalue.上页下页返圆
上页 下页 返回 This is a definite integral, we can just work out its value. tdt a c L a = − 2 0 2 2 2 4 1 cos According to the elliptical parametric equation and arc length formula of curve, we can calculate the perimeter of elliptical orbit
According to I = ['f(x)dx, if we can find out Integrand function f(x) and original function F(x), then we can work out thefollowing Newton-Leibniz formula[~ f(x)dx = F(b) - F(a)But considering of large number of integrand functions f(x)actually it's difficult in practical problems上页下页返圆
上页 下页 返回 f (x)dx F(b) F(a) b a = − According to , if we can find out Integrand function f (x) and original function F(x)， then we can work out the following Newton—Leibniz formula = b a I f(x )dx But considering of large number of integrand functions f (x), actually it’s difficult in practical problems
sinx(1 ) suppose that f(x) is, sin x?,We can not find out theYelementary function of original function( 2 ) When f (x) is based on an numerical measurements to calculatea sheet of data, we can not use Newton-Leibniz formula directly(3 ) When the structure of f(x) is complicated, we would findthat it's difficult to work out original function. At that time, it'snecessary to study integral numerical problem.上页下页返圆
上页 下页 返回 x sin x （1）suppose that f (x) is , sin x 2 ， We can not find out the elementary function of original function （2） When f (x) is based on an numerical measurements to calculate a sheet of data, we can not use Newton—Leibniz formula directly. （3） When the structure of f (x) is complicated, we would find that it’s difficult to work out original function. At that time, it’s necessary to study integral numerical problem