Question

USE PYTHON TO SOLVE

3) Given f(x) = e^-x

(a) Estimate the area under the graph between x = 0 and x = 2 with n = 10, 100, 1000 left endpoint rectangles.

(b) Compare your answers in part (a) to the exact area, 1 − (1 / e^2 ). What do you notice?

Answer 1

Python Script:

import math

def func(x):
return math.exp(-x)

N = [10, 100, 1000]
Sum = [0, 0, 0]
for i in range(len(N)):
n = N[i]
x = 0
h = 2/n
for j in range(n):
Sum[i] += func(x)*h
x += h
print('Area (n = ' + str(n) + ') =', Sum[i])
print('Exact Area =', 1 - 1/math.exp(2))

Output:

Area (n = 10) = 0.9540114845132726
Area (n = 100) = 0.873340185896101
Area (n = 1000) = 0.8655296697017035
Exact Area = 0.8646647167633873

Note: Approximation approaches the exact value as we increase the number of intervals.