Trapezoidal rule (trapezoid) in SciPy - Practice Problems & Coding Challenges

Challenge - 5 Problems

❓ Predict Output

intermediate

Output of trapezoidal integration on simple data

What is the output of the following code that uses scipy's trapezoidal rule to integrate y = [1, 2, 3, 4] over x = [0, 1, 2, 3]?

SciPy

import numpy as np
from scipy.integrate import trapezoid
x = np.array([0, 1, 2, 3])
y = np.array([1, 2, 3, 4])
result = trapezoid(y, x)
print(result)

A 7.5

B 10.0

C 9.0

D 8.5

💡 Hint

Remember trapezoidal rule sums areas of trapezoids between points.

The trapezoidal rule calculates the integral as the sum of trapezoid areas. Here, the intervals are 1 unit each, so the integral is (1+2)/2*1 + (2+3)/2*1 + (3+4)/2*1 = 1.5 + 2.5 + 3.5 = 7.5.

❓ data_output

intermediate

Number of intervals used in trapezoidal integration

Given x = [0, 2, 4, 6, 8] and y = [0, 4, 16, 36, 64], how many trapezoids does scipy.integrate.trapezoid use to compute the integral?

SciPy

import numpy as np
from scipy.integrate import trapezoid
x = np.array([0, 2, 4, 6, 8])
y = np.array([0, 4, 16, 36, 64])
result = trapezoid(y, x)
print(len(x) - 1)

A 4

B 5

C 1

D 3

💡 Hint

Number of trapezoids is one less than number of points.

The trapezoidal rule uses intervals between points. With 5 points, there are 4 intervals, so 4 trapezoids.

🔧 Debug

advanced

Identify the error in trapezoidal integration code

What error will this code raise? import numpy as np from scipy.integrate import trapezoid x = np.array([0, 1, 2]) y = np.array([1, 2]) result = trapezoid(y, x) print(result)

SciPy

import numpy as np
from scipy.integrate import trapezoid
x = np.array([0, 1, 2])
y = np.array([1, 2])
result = trapezoid(y, x)
print(result)

A No error, prints 1.5

B TypeError: unsupported operand type(s) for +: 'int' and 'str'

C IndexError: index out of range

D ValueError: x and y must have the same length

💡 Hint

Check if x and y arrays have the same length.

The trapezoid function requires x and y arrays to be the same length. Here, y has length 2 but x has length 3, so it raises a ValueError.

🚀 Application

advanced

Calculate integral of sin(x) from 0 to pi using trapezoidal rule

Using numpy and scipy, which option correctly computes the integral of sin(x) from 0 to π using 100 intervals?

SciPy

import numpy as np
from scipy.integrate import trapezoid
x = np.linspace(0, np.pi, 101)
y = np.sin(x)
result = trapezoid(y, x)
print(round(result, 4))

A 2.0000

B 1.9998

C 3.1416

D 1.0000

💡 Hint

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

Using 100 intervals, trapezoidal rule approximates the integral close to 2, here 1.9998 is the rounded result.

🧠 Conceptual

expert

Effect of non-uniform spacing on trapezoidal integration

Which statement best describes how scipy.integrate.trapezoid handles non-uniform spacing in x values?

A It averages all x values and uses that average as the spacing for all trapezoids

B It assumes uniform spacing and ignores x values, leading to incorrect results if spacing varies

C It uses the differences between consecutive x values to weight the trapezoids correctly

D It requires x values to be equally spaced or raises an error

💡 Hint

Think about how trapezoidal rule calculates area between points.

The trapezoidal rule calculates area using the width between points. scipy.integrate.trapezoid uses the actual spacing between x points, so it works correctly with non-uniform spacing.