1446.sagews

def right_sum(f, a, b, n):
    deltax = ((b-a)/n).n()
    return(sum(f(a + j*deltax) for j in [1..n])*deltax)

right_sum(sin,0,pi/2,4)

1.18346534182214

def left_sum(f, a, b, n):
    deltax = ((b-a)/n).n()
    return(sum(f(a + j*deltax) for j in [1..n-1])*deltax)

def midpoint_sum(f, a, b, n):
    deltax = ((b-a)/n).n()
    return(sum(f(a + (j+1/2)*deltax) for j in [0..n-1])*deltax)

midpoint_sum(sin,0,pi/2,10)

0.988704498205414

def trapezoid_sum(f, a, b, n):
    return((right_sum(f,a,b,n) + left_sum(f,a,b,n))/2)

trapezoid_sum(sin, 0, pi/2, 10)

0.997942986354357

right_sum(sin, 0,  pi/2, 1)

1.57079632679490

right_sum(sin, 0,  pi/2, 10)

1.07648280269410

right_sum(sin, 0,  pi/2, 100)

1.00783341987358

right_sum(sin, 0,  pi/2, 1000)

1.00078519254663

right_sum(sin, 0,  pi/2, 10000)

1.00007853776017

right_sum(sin, 0,  pi/2, 100000)

1.00000785396107

left_sum(sin, 0,  pi/2, 1)

0.000000000000000

left_sum(sin, 0,  pi/2, 10)

0.919403170014612

left_sum(sin, 0,  pi/2, 100)

0.992125456605633

left_sum(sin, 0,  pi/2, 1000)

0.999214396219836

left_sum(sin, 0,  pi/2, 10000)

0.999921458127493

left_sum(sin, 0,  pi/2, 100000)

0.999992145997806

1-trapezoid_sum(sin, 0, pi/2, 1)

0.214601836602552

1-trapezoid_sum(sin, 0, pi/2, 10)

0.00205701364564281

1-trapezoid_sum(sin, 0, pi/2, 100)

0.0000205617603923347

1-trapezoid_sum(sin, 0, pi/2, 1000)

2.05616766724148e-7

1-trapezoid_sum(sin, 0, pi/2, 10000)

2.05616734749725e-9

1-trapezoid_sum(sin, 0, pi/2, 100000)

2.05599981484283e-11

1-midpoint_sum(sin, 0, pi/2, 1)

1.00000000000000

1-midpoint_sum(sin, 0, pi/2, 10)

0.0112955017945863

1-midpoint_sum(sin, 0, pi/2, 100)

0.000113087874764717

1-midpoint_sum(sin, 0, pi/2, 1000)

1.13089203646055e-6

1-midpoint_sum(sin, 0, pi/2, 10000)

1.13089210218575e-8

1-midpoint_sum(sin, 0, pi/2, 100000)

1.13089426712065e-10

1-.919403

0.0805970000000000

1-.992125

0.00787499999999997

1-.99921

0.000789999999999957

1-.999921

0.0000790000000000513

1-.9999921

7.89999999994961e-6

plot(sin(x), x, -10, 10)

midpoint_sum(sin, 0, pi/2, 1)

0.000000000000000

def coop_sum(f, a, b, n):
    return((trapezoid_sum(f,a,b,n) + 2*midpoint_sum(f,a,b,n))/3)

coop_sum(sin,0,pi/2,1)

1.00227987749221

coop_sum(sin,0,pi/2,10)

1.00000021154659

coop_sum(sin,0,pi/2,100)

1.00000000002114

coop_sum(sin,0,pi/2,10000)

1.00000000000000

coop_sum((x^2), 0, 10, 1)

333.333333333333

coop_sum((x^2), 0, 10, 10)

333.333333333333