Why numerical integration computes areas in SciPy

Introduction

Numerical integration helps us find the area under a curve when we can't calculate it exactly. It adds up small slices to get the total area.

When you want to find the total distance traveled from speed data over time.

When you need to calculate the area under a graph that doesn't have a simple formula.

When you have experimental data points and want to estimate the total quantity represented by the curve.

When solving physics problems involving work done by a force over a distance.

When calculating probabilities from continuous probability distributions.

Syntax

SciPy

from scipy.integrate import quad

result, error = quad(function, lower_limit, upper_limit)

quad is a common function in SciPy for numerical integration.

You provide the function to integrate and the limits between which to find the area.

Examples

This calculates the area under the curve y = x² from 0 to 1.

SciPy

from scipy.integrate import quad

def f(x):
    return x**2

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

Using a lambda function to find the area under y = 3x + 2 from 1 to 4.

SciPy

from scipy.integrate import quad

area, error = quad(lambda x: 3*x + 2, 1, 4)
print(area)

Sample Program

This program calculates the total distance traveled by integrating the speed function over time from 0 to 3 seconds.

SciPy

from scipy.integrate import quad

def speed(t):
    return 3*t**2 + 2*t + 1

# Calculate distance traveled from time 0 to 3
distance, error = quad(speed, 0, 3)
print(f"Distance traveled: {distance:.2f}")

Important Notes

Numerical integration approximates the area by summing many small slices under the curve.

The smaller the slices, the more accurate the result, but it takes more computing power.

Functions passed to quad must be able to accept a number and return a number.

Summary

Numerical integration finds the area under curves by adding up small parts.

It is useful when exact formulas are hard or impossible to use.

SciPy's quad function makes numerical integration easy in Python.