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approxint1

approxint1 - INTEGRATION > f1:=sqrt(1-x/x 1-x f1:= x >...

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INTEGRATION > f1:=sqrt(1-x)/x; f1 := 1 - x x > int(f1,x); 2 1 - x + ln 1 - x - 1 ( ) - ln 1 - x + 1 ( ) > f2:=tan(x)^3*sec(x)^11; f2 := tan x ( ) 3 sec x ( ) 11 > g2:=int(f2,x); g2 := sin x ( ) 4 13 cos x ( ) 13 + 9 sin x ( ) 4 143 cos x ( ) 11 + 7 sin x ( ) 4 143 cos x ( ) 9 + 5 sin x ( ) 4 143 cos x ( ) 7 + 3 sin x ( ) 4 143 cos x ( ) 5 + sin x ( ) 4 143 cos x ( ) 3 - sin x ( ) 4 143 cos x ( ) - 1 143 sin x ( ) 2 cos x ( ) - 2 143 cos x ( ) > h2:= sec(x)^13/13-sec(x)^11/11; h2 := 1 13 sec x ( ) 13 - 1 11 sec x ( ) 11 > simplify(h2-g2); 0 APPROXIMATE INTEGRATION (1) Left endpoint and Right endpoint approximation > DELTAX:=(a,b,n)->(b-a)/n; DELTAX := a , b , n ( ) b - a n > XK:=(a,b,n,k)->a + DELTAX(a,b,n)*k; XK := a , b , n , k ( ) a + DELTAX a , b , n ( ) k > LEFTEND:=(f,a,b,n)->DELTAX(a,b,n)*add(f(XK(a,b,n,k)),k=0..n-1); LEFTEND := f , a , b , n ( ) DELTAX a , b , n ( ) add f XK a , b , n , k ( ) ( ), k = 0 .. n - 1 ( ) ( ) > ERRORLEFT:=(f,a,b,n)->evalf( int(f(x),x=a..b) - LEFTEND(f,a,b,n)); ERRORLEFT := f , a , b , n ( ) evalf a b f x ( ) dx - LEFTEND f , a , b , n ( )

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> RIGHTEND:=(f,a,b,n)->DELTAX(a,b,n)*add(f(XK(a,b,n,k)),k=1..n); RIGHTEND := f , a , b , n ( ) DELTAX a , b , n ( ) add f XK a , b , n , k ( ) ( ), k = 1 .. n ( ) ( ) > ERRORRIGHT:=(f,a,b,n)->evalf( int(f(x),x=a..b) - RIGHTEND(f,a,b,n)); ERRORRIGHT := f , a , b , n ( ) evalf a b f x ( ) dx - RIGHTEND f , a , b , n ( ) EXAMPLE > f:=x->1/x; f := x 1 x > f(x); 1 x > f(2); 1 2 > plot(f(x),x=1..2);
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