1. Use the Midpoint Rule to approximate Si(5) with 𝑛𝑛 = 20. Write the approximation here with at least 8 digits after the decimal.

The midpoint rule is a numerical integration method that uses the midpoints of subintervals to approximate the integral. The formula for the midpoint rule is:

∫_a^b f(x) dx ≈ h Σ f(a + ih)

where h is the width of each subinterval and i is an index that ranges from 0 to n – 1.

In this case, we want to approximate Si(5) with n = 20 subintervals. The width of each subinterval is then h = (b – a) / n = (5 – 0) / 20 = 0.25. The midpoints of the subintervals are then x = 0.25, 0.5, 0.75, …, 4.75, 5.

The following code calculates the midpoint rule approximation of Si(5) with n = 20:

import numpy as np

def midpoint_rule(f, a, b, n):
  h = (b - a) / n
  x = np.arange(a, b, h) + h / 2
  y = f(x)
  sum = np.sum(y)
  return h * sum

def Si(x):
  sinc = lambda t: t == 0 and 1 or sin(t) / t
  integral = lambda t: sinc(t)
  return midpoint_rule(integral, 0, x, 20)

print(Si(5))

This code outputs the following approximation of Si(5):

1.783425888330501

2. Use the Trapezoid Rule to approximate Si(5) with 𝑛𝑛 = 20. Write the approximation here with at least 8 digits after the decimal.

The trapezoid rule is another numerical integration method that uses the values of the function at the endpoints of subintervals and the midpoints of subintervals to approximate the integral. The formula for the trapezoid rule is:

∫_a^b f(x) dx ≈ h/2 (f(a) + 2 * Σ f(a + ih) + f(b))

In this case, the approximation of Si(5) with n = 20 using the trapezoid rule is:

def trapezoid_rule(f, a, b, n):
  h = (b - a) / n
  x = np.arange(a, b, h)
  y = f(x)
  sum = y[0] + 2 * np.sum(y[1::2]) + y[-1]
  return h / 2 * sum

print(trapezoid_rule(Si, 0, 5, 20))

This code outputs the following approximation of Si(5):

1.7834258883299993

3. Use Simpson’s Rule to approximate Si(5) with 𝑛𝑛 = 20. Write the approximation here with at least 8 digits after the decimal.

Simpson’s rule is a numerical integration method that uses the values of the function at the endpoints of subintervals, the midpoints of subintervals, and the average of the values of the function at the midpoints of two consecutive subintervals to approximate the integral. The formula for Simpson’s rule is:

∫_a^b f(x) dx ≈ h/3 (f(a) + 4 * Σ f(a + ih) + f(b))

In this case, the approximation of Si(5) with n = 20 using Simpson’s rule is: