Single integral (quad) in SciPy - Step-by-Step Execution

Concept Flow - Single integral (quad)

Define function f(x)

↓

Set integration limits a, b

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Call quad(f, a, b)

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quad computes integral approx

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Return integral value and error estimate

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Use or display result

We define the function to integrate, set limits, call quad to compute the integral, then get the result and error.

Execution Sample

SciPy

from scipy.integrate import quad

def f(x):
    return x**2

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

This code calculates the integral of x squared from 0 to 2 using quad.

Execution Table

Step	Action	Function Evaluation	Intermediate Result	Notes
1	Define function f(x) = x^2	N/A	Function ready	Function set for integration
2	Set limits a=0, b=2	N/A	Limits set	Integration bounds defined
3	Call quad(f, 0, 2)	quad evaluates f at sample points	Approximate integral computed	Adaptive quadrature starts
4	quad returns result and error	N/A	Result ~2.6667, error ~3e-14	Integral value and error estimate
5	Print result	N/A	2.6666666666666665	Output integral value

💡 quad finishes when desired accuracy is reached or max iterations done

Variable Tracker

Variable	Start	After quad call	Final
f	Defined as x^2	Used internally by quad	No change
a	0	0	0
b	2	2	2
result	Undefined	2.6666666666666665	2.6666666666666665
error	Undefined	3.0e-14	3.0e-14

Key Moments - 3 Insights

Why does quad return two values, not just the integral?

quad returns both the integral estimate and an error estimate, as shown in execution_table step 4, to tell you how accurate the result is.

What if the function is not defined at some points inside the limits?

quad samples points adaptively; if the function is undefined or has issues, quad may raise an error or give a warning. This is why function definition matters (step 1).

Why do we need to define the function separately before calling quad?

quad needs a callable function to evaluate at many points inside [a,b]. Defining f(x) first (step 1) lets quad call it repeatedly.

Visual Quiz - 3 Questions

Look at the execution_table at step 4, what does quad return?

A Integral value and error estimate

B Only the error estimate

C Only the integral value

D The function definition

💡 Hint

Check the 'Intermediate Result' column at step 4 in execution_table

According to variable_tracker, what is the value of 'result' after quad finishes?

A Undefined

B 0

C 2.6666666666666665

D 3.0e-14

💡 Hint

Look at the 'result' row under 'After quad call' in variable_tracker

If we change the upper limit b from 2 to 3, what will happen in the execution_table?

A The integral value will stay the same

B The integral value will likely increase

C quad will return an error

D The function f(x) will change

💡 Hint

Integral over a larger interval usually changes the result, see concept_flow

Concept Snapshot

Use scipy.integrate.quad to compute single integrals.
Define the function f(x) first.
Call quad(f, a, b) with limits a and b.
quad returns (integral, error estimate).
Use the integral value as the result.
Error helps check accuracy.

Full Transcript

This visual execution shows how to compute a single integral using scipy's quad function. First, we define the function to integrate, here f(x) = x squared. Then we set the integration limits from 0 to 2. Calling quad with these inputs makes it evaluate the function at many points and approximate the integral. The output is two values: the integral estimate and an error estimate. We track variables like the function, limits, result, and error through the steps. Key moments clarify why quad returns two values and why the function must be defined first. The quiz tests understanding of quad's output and variable values. The snapshot summarizes the process simply.