1148.sagews

f(x) = ln(x); f;numerical_integral(f, 1, 5)

x |--> log(x) (4.0471895621705025, 4.4932830369442986e-14)

def rsum_lh(a, b, n):
    Dx = (b-a)/n
    return RR(sum([f(a+i*Dx)*Dx for i in range(n)]))

rsum_lh(1, 5, 10)

3.71470292908879

rsum_lh(1, 5, 50)

3.98238549166703

rsum_lh(1, 5, 100)

4.01489414430613

def rsum_rh(a, b, n):
    Dx = (b-a)/n
    return RR(sum([f(a+(i+1)*Dx)*Dx for i in range(n)]))

rsum_rh(1, 5, 10)

4.35847809406243

rsum_rh(1, 5, 20)

4.20547104630329

rsum_rh(1, 5, 50)

4.11114052466176

plot(ln(x), (0,5))

g(x) = sqrt(1 + x^4); g

x |--> sqrt(x^4 + 1)

f(x) = diff(g, x, 4); f

x |--> -240*x^12/(x^4 + 1)^(7/2) + 432*x^8/(x^4 + 1)^(5/2) - 204*x^4/(x^4 + 1)^(3/2) + 12/sqrt(x^4 + 1)

plot(diff(g, x, 4), (-5,5))

def rsum_lh(a, b, n):
    Dx = (b-a)/n
    return RR(sum([f(a+i*Dx)*Dx for i in range(n)]))

rsum_lh(0, 5, 1)

60.0000000000000

rsum_lh(0, 5, 10)

3.57792510642994

rsum_lh(0, 5, 100)

0.296085456001272

rsum_lh(0, 5, 500)

0.0561623914891393

def rsum_rh(a, b, n):
    Dx = (b-a)/n
    return RR(sum([f(a+(i+1)*Dx)*Dx for i in range(n)]))

rsum_rh(0, 5, 1)

0.0190073396756721

rsum_rh(0, 5, 10)

-2.42017415960250

rsum_rh(0, 5, 100)

-0.303724470601971

rsum_rh(0, 5, 500)

-0.0637995938315088

def trapezoidal_rule(a, b, n):
    Deltax = (b-a)*1.0/n
    coeffs = [2]*(n-1)
    coeffs = [1]+coeffs+[1]
    valsf = [f(a+Deltax*i) for i in range(n+1)]
    return RR((Deltax/2)*sum([coeffs[i]*valsf[i] for i in range(n+1)]))

trapezoidal_rule(0, 5, 1)

30.0095036698378

trapezoidal_rule(0, 5, 10)

0.578875473413701

trapezoidal_rule(0, 5, 100)

-0.00381950730034719

trapezoidal_rule(0, 5, 500)

-0.00381860117118486

def simpsons_rule(a,b,n):
    Deltax = (b-a)*1.0/n
    n2 = int(n/2)
    coeffs = [4,2]*n2
    coeffs = [1]+coeffs[:n-1]+[1]
    valsf = [f(a+Deltax*i) for i in range(n+1)]
    return RR((Deltax/3)*sum([coeffs[i]*valsf[i] for i in range(n+1)]))

simpsons_rule(0, 5, 1)

20.0063357798919

simpsons_rule(0, 5, 10)

1.36076980408666

simpsons_rule(0, 5, 100)

-0.00381856375637102

simpsons_rule(0, 5, 500)

-0.00381856341290844

r(x) = sqrt(1 + x^4); r

x |--> sqrt(x^4 + 1)

s(x) = diff(r, x, 4); s

x |--> -240*x^12/(x^4 + 1)^(7/2) + 432*x^8/(x^4 + 1)^(5/2) - 204*x^4/(x^4 + 1)^(3/2) + 12/sqrt(x^4 + 1)

RR(integral(s, 0, 5))

-0.00381856341235940

h(x) = tan(x) - sec(x); h

x |--> tan(x) - sec(x)

t(x) = diff(h); t

x |--> tan(x)^2 - tan(x)*sec(x) + 1

assume(cos(x)>0)
RR(integral(t, 0, pi/4))

0.585786437626905

def rsum_lh(a, b, n):
    Dx = (b-a)/n
    return RR(sum([t(a+i*Dx)*Dx for i in range(n)]))

rsum_lh(0, pi/4, 500)

0.586111915923777

def rsum_rh(a, b, n):
    Dx = (b-a)/n
    return RR(sum([t(a+(i+1)*Dx)*Dx for i in range(n)]))

rsum_rh(0, pi/4, 500)

0.585461270781493

def trapezoidal_rule(a, b, n):
    Deltax = (b-a)*1.0/n
    coeffs = [2]*(n-1)
    coeffs = [1]+coeffs+[1]
    valsf = [t(a+Deltax*i) for i in range(n+1)]
    return RR((Deltax/2)*sum([coeffs[i]*valsf[i] for i in range(n+1)]))

trapezoidal_rule(0, pi/4, 1)

0.622736877826377

trapezoidal_rule(0, pi/4, 10)

0.586175520496119

trapezoidal_rule(0, pi/4, 500)

0.585786593352635

def simpsons_rule(a,b,n):
    Deltax = (b-a)*1.0/n
    n2 = int(n/2)
    coeffs = [4,2]*n2
    coeffs = [1]+coeffs[:n-1]+[1]
    valsf = [t(a+Deltax*i) for i in range(n+1)]
    return RR((Deltax/3)*sum([coeffs[i]*valsf[i] for i in range(n+1)]))

simpsons_rule(0, pi/4, 1)

0.415157918550918

simpsons_rule(0, pi/4, 10)

0.585787356800239

simpsons_rule(0, pi/4, 500)

0.585786437627051