Single integral (quad) in SciPy - Practice Problems & Coding Challenges

Challenge - 5 Problems

❓ Predict Output

intermediate

Output of a definite integral using scipy.quad

What is the output of this code that calculates the integral of f(x) = x^2 from 0 to 2 using scipy.quad?

SciPy

from scipy.integrate import quad

def f(x):
    return x**2

result, error = quad(f, 0, 2)
print(round(result, 3))

A 4.000

B 2.667

C 1.333

D 8.000

💡 Hint

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

The integral of x^2 from 0 to 2 is (2^3)/3 = 8/3 ≈ 2.667. The code uses scipy.quad to compute this numerically.

❓ data_output

intermediate

Number of points used by scipy.quad for integral approximation

What is the type and approximate number of points used internally by scipy.quad when integrating f(x) = sin(x) from 0 to π?

SciPy

from scipy.integrate import quad
import numpy as np

def f(x):
    return np.sin(x)

result, error = quad(f, 0, np.pi, full_output=1)
print(result)
print(error['neval'])

A Result: 0.0, Points used: around 10

B Result: 1.0, Points used: around 5

C Result: 2.0, Points used: around 50

D Result: 3.1415, Points used: around 100

💡 Hint

The integral of sin(x) from 0 to π is 2.

The integral of sin(x) from 0 to π is exactly 2. scipy.quad uses adaptive quadrature and typically evaluates around 50 points for this integral.

🧠 Conceptual

advanced

Understanding error estimate in scipy.quad

What does the second value returned by scipy.integrate.quad represent when computing an integral?

A The number of function evaluations performed

B The time taken to compute the integral

C The relative error between two integration methods

D The estimated absolute error of the integral approximation

💡 Hint

Think about what error means in numerical integration.

scipy.quad returns two values: the integral result and an estimate of the absolute error in the result.

🔧 Debug

advanced

Identify the error in this scipy.quad usage

What error will this code raise when trying to integrate f(x) = 1/x from 0 to 1 using scipy.quad?

SciPy

from scipy.integrate import quad

def f(x):
    return 1/x

result, error = quad(f, 0, 1)
print(result)

A No error, returns a finite result

B ValueError: Integration bounds must be finite

C RuntimeWarning: divide by zero encountered in true_divide

D ZeroDivisionError

💡 Hint

Consider what happens at x=0 for 1/x.

The function 1/x has an improper integral at x=0 but scipy.quad can handle improper integrals with singularities at the boundary and returns a finite result.

🚀 Application

expert

Calculate integral of a piecewise function using scipy.quad

Using scipy.quad, what is the value of the integral of the function f(x) defined as f(x) = x for x < 1 and f(x) = 2 - x for x ≥ 1, integrated from 0 to 2?

SciPy

from scipy.integrate import quad

def f(x):
    return x if x < 1 else 2 - x

result, error = quad(f, 0, 2)
print(round(result, 3))

A 1.0

B 2.0

C 0.5

D 1.5

💡 Hint

Split the integral at x=1 and sum the two parts.

The integral from 0 to 1 of x dx is 0.5, and from 1 to 2 of (2 - x) dx is also 0.5, total 1.0.