Numerical Integrals Various Rules

## Description

This interact estimates the integral of a function using Riemann Sums under different rules. The user may choose the function, the interval of integration, the number of Riemann estimates and the rule to use for area estimation.

## Sage Cell

#### Code

var('x')
@interact
def midpoint(f = input_box(default = sin(x^2) + 2, type = SR),
interval=range_slider(0, 10, 1, default=(0, 4), label="Interval"),
number_of_subdivisions = slider(1, 20, 1, default=4, label="Number of boxes"),
endpoint_rule = selector(['Midpoint', 'Left', 'Right', 'Upper', 'Lower'], nrows=1, label="Endpoint rule")):

a, b = map(QQ, interval)
t = var('t')
func = fast_callable(f(x=t), RDF, vars=[t])
dx = ZZ(b-a)/ZZ(number_of_subdivisions)

xs = []
ys = []
for q in range(number_of_subdivisions):
if endpoint_rule == 'Left':
xs.append(q*dx + a)
elif endpoint_rule == 'Midpoint':
xs.append(q*dx + a + dx/2)
elif endpoint_rule == 'Right':
xs.append(q*dx + a + dx)
elif endpoint_rule == 'Upper':
x = find_local_maximum(func, q*dx + a, q*dx + dx + a)
xs.append(x)
elif endpoint_rule == 'Lower':
x = find_local_minimum(func, q*dx + a, q*dx + dx + a)
xs.append(x)
ys = [ func(x) for x in xs ]

rects = Graphics()
for q in range(number_of_subdivisions):
xm = q*dx + dx/2 + a
x = xs[q]
y = ys[q]
rects += line([[xm-dx/2,0],[xm-dx/2,y],[xm+dx/2,y],[xm+dx/2,0]], rgbcolor = (1,0,0))
rects += point((x, y), rgbcolor = (1,0,0))
min_y = min(0, find_local_minimum(func,a,b))
max_y = max(0, find_local_maximum(func,a,b))

# html('<h3>Numerical integrals with the midpoint rule</h3>')
show(plot(func,a,b) + rects, xmin = a, xmax = b, ymin = min_y, ymax = max_y)

def cap(x):
# print only a few digits of precision
if x < 1e-4:
return 0
return RealField(20)(x)
sum_html = "%s \\cdot \\left[ %s \\right]" % (dx, ' + '.join([ "f(%s)" % cap(i) for i in xs ]))
num_html = "%s \\cdot \\left[ %s \\right]" % (dx, ' + '.join([ str(cap(i)) for i in ys ]))

estimated_answer = dx * sum([ ys[q] for q in range(number_of_subdivisions)])

pretty_print(html(r'''
<div class="math">
\begin{align*}
\int_{a}^{b} {f(x) \, dx} & = %s \\\
\sum_{i=1}^{%s} {f(x_i) \, \Delta x} & = %s \\\
& = %s \\\
& = %s . \end{align*} </div>'''
% (numerical_answer, number_of_subdivisions, sum_html, num_html, estimated_answer)))


none

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