Romberg integration in SciPy - Practice Problems & Coding Challenges

Challenge - 5 Problems

❓ Predict Output

intermediate

Output of Romberg integration for a simple function

What is the output of the following code that uses Romberg integration to calculate the integral of f(x) = x^2 from 0 to 1?

SciPy

from scipy.integrate import romberg

f = lambda x: x**2
result = romberg(f, 0, 1, divmax=5)
print(round(result, 5))

A 0.33333

B 0.50000

C 0.25000

D 1.00000

💡 Hint

Recall the integral of x^2 from 0 to 1 is 1/3.

The integral of x^2 from 0 to 1 is 1/3, which is approximately 0.33333. Romberg integration computes this accurately.

❓ data_output

intermediate

Number of iterations in Romberg integration

How many iterations does Romberg integration perform with divmax=3 when integrating f(x) = sin(x) from 0 to pi?

SciPy

from scipy.integrate import romberg
import numpy as np

f = np.sin
result = romberg(f, 0, np.pi, divmax=3, show=True)

A 3

B 4

C 2

D 5

💡 Hint

Romberg integration performs divmax+1 iterations.

Romberg integration performs divmax+1 iterations, so with divmax=3, it performs 4 iterations.

🔧 Debug

advanced

Identify the error in Romberg integration usage

What error will this code raise when trying to compute the integral of f(x) = 1/x from 0 to 1 using Romberg integration?

SciPy

from scipy.integrate import romberg

f = lambda x: 1/x
result = romberg(f, 0, 1)

A TypeError

B ValueError

C No error, returns a float

D ZeroDivisionError

💡 Hint

Check the function behavior at the lower limit 0.

The function 1/x is undefined at x=0, causing a ZeroDivisionError during evaluation.

🧠 Conceptual

advanced

Understanding Romberg integration accuracy

Which statement best describes why Romberg integration is more accurate than the trapezoidal rule alone?

A It applies Simpson's rule repeatedly for better accuracy.

B It uses Monte Carlo sampling to reduce error.

C It uses Richardson extrapolation to improve trapezoidal estimates.

D It increases the number of trapezoids without combining results.

💡 Hint

Romberg integration builds on trapezoidal rule results.

Romberg integration applies Richardson extrapolation to trapezoidal rule results to accelerate convergence and improve accuracy.

🚀 Application

expert

Using Romberg integration for a complex function

You want to compute the integral of f(x) = exp(-x^2) from -2 to 2 using Romberg integration. Which code snippet correctly computes this and rounds the result to 6 decimals?

A

from scipy.integrate import romberg
import numpy as np
f = lambda x: np.exp(-x**2)
result = romberg(f, -2, 2)
print(round(result, 6))

B

from scipy.integrate import romberg
import math
f = lambda x: math.exp(-x**2)
result = romberg(f, 0, 2)
print(round(result, 6))

C

from scipy.integrate import romberg
import numpy as np
f = lambda x: np.exp(-x**2)
result = romberg(f, 0, 2)
print(round(result, 6))

D

from scipy.integrate import romberg
import math
f = lambda x: math.exp(-x**2)
result = romberg(f, -2, 2)
print(round(result, 6))

💡 Hint

Use numpy for vectorized math and correct integration limits.

Option A correctly imports numpy for the exponential function, uses the full interval from -2 to 2, and rounds the result. Options B and C use wrong limits, and A uses math.exp which may cause issues with array inputs.