404-Rectangle Rule.sagews

Trapezoidal Rule

Approximation of $\int_a^b f(x) {\rm d}x$ for $x \in [a, b]$ using the trapezoid rule.

$$\sum_{i=0}^{n-1} \frac{f(x_i)+f(x_i+\Delta_x)}{2} \cdot \Delta_x$$

for $x_i = a + i \cdot \Delta_x$ with $\Delta_x = \frac{b-a}{n}$ and $n \in \mathbb{N}^+$

%auto

# the function we are integrating
var('x')
f(x) = 1/2 * x^2 - 2*x + 1

def next_hue(dx):
    """
    generator object for repetitive calls in the rgbcolor argument
    """
    x = 0
    while 1:
        if x > 1: x = 0
        yield hue(x, 0.3)
        x += dx

%auto
@interact
def _(ab=range_slider(-4,10,0.1,(-2,6),'[a, b]'), dx=slider(0.1,3,0.1,1,'dx')):
    a, b = ab
    xvals = srange(a,b,dx, include_endpoint=True)
    fx = [f(x) for x in xvals]
    fg = plot(f, (x,a,b))
    pts = point2d(zip(xvals,fx), rgbcolor='#ff3030', pointsize=30)
    lines = [([i,0], [i,f(i)], [i+dx, f(i+dx)], [i+dx, 0]) for i in xvals[0:-1]]
    nh = next_hue(1.0/len(xvals))
    areas = sum(map(lambda p : polygon(p, rgbcolor=nh.next()), lines))
    show(fg + areas + pts)
    areas = [  ( f(i) + f(i+dx) )/2 for i in xvals[0:-1]]
    print 'approximation: %s' % float(dx * sum(areas))
    print 'numerical_integral: %s'  % numerical_integral(f, a, b)[0]

Rectangle Rule

Approximation of $\int_a^b f(x) {\rm d} x$ with $x \in [a,b]$ with

$\frac{f(a) \cdot  \Delta_x}{2}  + \sum_{i=1}^{n-1} f(x_i) \cdot \Delta_x  + \frac{f(b) \cdot  \Delta_x}{2}$

for $x_i = a+ i \cdot \Delta_x$ with $\Delta_x = \frac{b-1}{n}$ and $n \in \mathbb{N}^+$

%auto
@interact
def _(ab=range_slider(-4,10,0.1,(-2,6),'[a, b]'), dx=slider(0.1,3,0.1,0.5,'dx')):
    a, b = ab
    xvals = srange(a,b,dx, include_endpoint=True)
    fx = [f(x) for x in xvals]
    fg = plot(f, (x,a,b))
    pts = point2d(zip(xvals,fx), rgbcolor='#ff3030', pointsize=30)
    #lines = [([i,0], [i,f(i)], [i+dx, f(i+dx)], [i+dx, 0]) for i in xvals[0:-1]]
    lines = [([xvals[0], 0], [xvals[0], f(xvals[0])], [xvals[0] + dx/2, f(xvals[0])], [xvals[0] + dx/2, 0])]
    for i in xvals[1:-1]:
        x1, x2 = i-dx/2, i+dx/2
        lines.append(([x1,0], [x1, f(i)], [x2, f(i)], [x2, 0]))
    x1, x2 = xvals[-1]-dx/2, xvals[-1]
    lines.append(([x1,0], [x1, f(x2)], [x2, f(x2)], [x2, 0]))
    nh = next_hue(1.0/len(xvals))
    areas = sum(map(lambda p : polygon(p, rgbcolor=nh.next()), lines))
    show(fg + areas + pts)
    areas = [f(i) for i in xvals[1:-1]]
    areas.append(f(xvals[0]) / 2)
    areas.append(f(xvals[-1]) / 2)
    print 'approximation: %s ' % float(dx * sum(areas))
    print 'numerical_integral: %s'  % numerical_integral(f, a, b)[0]