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C PROGRAM : USE TRAPEZOIDAL RULE FOR EVALUATING

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C PROGRAM : USE TRAPEZOIDAL RULE FOR EVALUATING

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  1. PROGRAM 8consider a function y = f(x). In order to evaluate definite integral I = a∫b y dx =a∫b f(x) dxDivide the interval (b-a) of x into n equal parts starting from x0=ai.e. xo , (x0+h), (x0+2h),…..,(x0+nh) .Width of each part =h = (b-a)/n The values of function y = f(x) at xo , (x0+h), (x0+2h),…..,(x0+nh) arey0,y1,y2,……yn respectively. Then using Newton’s formula I = a∫b f(x) dx = x0∫xo+nh f(x) dxThen using Newton’s formula,s I = h[ny0 + (n2/2) ∆y0 + { (n3/3 ) – (n2/2)}(∆2 yo/2!) + ………] ……..(1) Equation (1) is known as general quadrature formula .TRAPEZOIDAL RULE :Taking n = 1 and neglecting second and higher order termsx0∫x0+h y dx = h [ ( y0+ ∆y0 /2) ] = h [ y0 + (y1-y0 )/2 ] = (h/2) [ y0 + y1 ] Similarly for the next intervals,x0+h ∫ x0+2h y dx = (h/2) [ y1 + y2 ]. .x0+(n-1) h ∫ x0+nh y dx = (h/2) [ yn-1 + yn ]

  2. On adding all these terms,(i.e. by Trapezoidal Rule the value of definite or finite integral is given by I = a∫b y dx = (h/2) [ ( y0+yn) + 2( y1 + y2 +….+yn-1 )] I = (h/2) [ (sum of first and last terms ) + 2 (sum of remaining terms )] where y0 =F(xo) = F (a) y1 =F(xo + H) = F (a + H) y2=F(xo + 2H) = F (a +2 H) . . yn =F(xo + nH) = F (a +n H) =F (b)

  3. C PROGRAM : USE TRAPEZOIDAL RULE FOR EVALUATING DEFINITE INTEGRAL OF FUNCTION F=1-EXP(-X/2.0) C MAIN PROGRAM WRITE(*,*)’GIVE INITIAL AND FINAL VALUES OF X’ READ(*,*) A , B WRITE(*,*)’GIVE THE SEGMENT WIDTH’ READ(*,*) H N=(B-A)/H SUM=(F(A)+F(B))/ 2.0 DO 10 I=1,N-1 SUM= SUM+F(A+I*H) 10 CONTINUE RESULT=SUM*H WRITE(*,*)’INTEGRAL BETWEEN’,A,’AND’,B WRITE(*,*)’WHEN H =‘, H , ‘IS’, RESULT STOP END C ……END OF MAIN PROGRAM…….

  4. FUNCTION SUBPROGRAM FUNCTION F(X) F=1-EXP(-X/2.0) RETURN END

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