Numerical integrals with the midpoint rule

## Description

This Sage interact allows the user to numerically approximate

(1)\begin{align} \int_a^b f(x) \, dx \end{align}

using a midpoint approximation. The user may input a function, upper and lower limits, and the number of rectangles used in the approximation.

## Sage Cell

#### Code

```
var('x')
@interact
def midpoint(n = slider(1,100,1,4), f = input_box(default = "x^2", type = str), start = input_box(default = "0", type = str), end = input_box(default = "1", type = str)):
a = N(start)
b = N(end)
func = sage_eval(f, locals={'x':x})
dx = (b-a)/n
midxs = [q*dx+dx/2 + a for q in range(n)]
midys = [func(x=x_val) for x_val in midxs]
rects = Graphics()
for q in range(n):
xm = midxs[q]
ym = midys[q]
rects = rects + line([[xm-dx/2,0],[xm-dx/2,ym],[xm+dx/2,ym],[xm+dx/2,0]], rgbcolor = (1,0,0)) + point((xm,ym), rgbcolor = (1,0,0))
min_y = min(0, find_local_minimum(func,a,b)[0])
max_y = max(0, find_local_maximum(func,a,b)[0])
pretty_print(html('<h3>Numerical integrals with the midpoint rule</h3>'))
pretty_print(html(r'$\int_{a}^{b}{f(x) dx} {\approx} \sum_i{f(x_i) \Delta x}$'))
print("\n\nSage numerical answer: " + str(integral_numerical(func,a,b,max_points = 200)[0]))
print("Midpoint estimated answer: " + str(RDF(dx*sum([midys[q] for q in range(n)]))))
show(plot(func,a,b) + rects, xmin = a, xmax = b, ymin = min_y, ymax = max_y)
```

## Options

None.

## Tags

Primary Tags: Single variable calculus: Integration

Secondary Tags: Integration: Approximation

## Related Cells

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## Attribute

Permalink: https://wiki.sagemath.org/interact/calculus#Numerical_integrals_with_the_midpoint_rule

Author: Marshall Hampton

Date: 01 Jul 2020 21:33

Submitted by: Tom Judson