Question

4. Another approximation for integrals is the Trapezoid Rule:

integral (a to b)f(x) dx ≈ ∆x/2 (f(x_0) + 2f(x_1) + 2f(x_2) + · · · + 2f(x_n−2) + 2f(x_(n−1)) + f(x_n))

There is a built-in function trapz in the package scipy.integrate (refer to the Overview for importing and using this and the next command).

(a) Compute the Trapezoid approximation using n = 100 subintervals.

(b) Is the Trapezoid approximation equal to the average of the Left and Right Endpoint approximations? (c) Run the following code to illustrate the trapezoid method with 4 trapezoids (make sure you imported sympy as sp as stated in the Overview):

• x=sp.symbols(’x’)

• f=sp.exp(x/2)/x**3

• sp.plot(f,(x,1,5))

• xp=[1,2,3,4,5]

• yp=[f.subs({x:i}) for i in xp]

• import matplotlib.pyplot as plt

• plt.plot(xp,yp) Notice that the trapezoid approximation is obtained by using lines to estimate f(x) on each subinterval.

Answer 1

Code

import math
import matplotlib.pyplot as plt
import numpy as np


def trapz():
    a = 1
    b = 5
    N = 100
    h = (b - a) / N
    x = np.linspace(a, b, N)
    f = lambda x: math.exp(x / 2) / x ** 3
    y = []
    summation = 0
    for i in range(len(x)):
        x_val = f(x[i - 1]) + f(x[i])
        summation = summation + x_val * h * 0.5
        y.append(summation)
    print(summation)
    plt.plot(x, y)
    plt.show()


trapz()

# please comment in case of any query

Output