Trapezoidal rule (trapezoid) in SciPy - Step-by-Step Execution

Concept Flow - Trapezoidal rule (trapezoid)

Start with data points x and y

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Calculate differences between x points (dx)

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Calculate average of adjacent y values

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Multiply dx by average y values

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Sum all these products

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Result: Approximate integral using trapezoidal rule

The trapezoidal rule approximates the area under a curve by summing trapezoids formed between data points.

Execution Sample

SciPy

import numpy as np
from scipy.integrate import trapezoid

x = np.array([0, 1, 2, 3])
y = np.array([0, 1, 4, 9])
area = trapezoid(y, x)
print(area)

This code calculates the approximate integral of y with respect to x using the trapezoidal rule.

Execution Table

Step	Action	Values	Result
1	Input arrays x and y	x=[0,1,2,3], y=[0,1,4,9]	Ready for calculation
2	Calculate dx (differences between x)	dx=[1,1,1]	dx computed
3	Calculate average of adjacent y values	avg_y=[(0+1)/2, (1+4)/2, (4+9)/2] = [0.5, 2.5, 6.5]	avg_y computed
4	Multiply dx by avg_y element-wise	products=[1*0.5, 1*2.5, 1*6.5] = [0.5, 2.5, 6.5]	products computed
5	Sum all products	sum=0.5+2.5+6.5=9.5	Approximate integral = 9.5
6	Print result	area=9.5	Output: 9.5

💡 All trapezoids processed, sum gives approximate integral

Variable Tracker

Variable	Start	After Step 2	After Step 3	After Step 4	Final
x	[0,1,2,3]	[0,1,2,3]	[0,1,2,3]	[0,1,2,3]	[0,1,2,3]
y	[0,1,4,9]	[0,1,4,9]	[0,1,4,9]	[0,1,4,9]	[0,1,4,9]
dx	N/A	[1,1,1]	[1,1,1]	[1,1,1]	[1,1,1]
avg_y	N/A	N/A	[0.5,2.5,6.5]	[0.5,2.5,6.5]	[0.5,2.5,6.5]
products	N/A	N/A	N/A	[0.5,2.5,6.5]	[0.5,2.5,6.5]
area	N/A	N/A	N/A	N/A	9.5

Key Moments - 3 Insights

Why do we take the average of adjacent y values?

Because each trapezoid's height is the average of the two y-values at its base, as shown in step 3 of the execution_table.

Why do we multiply dx by the average y values?

Multiplying dx (width) by average y (height) gives the area of each trapezoid, as shown in step 4 of the execution_table.

What happens if x values are not equally spaced?

The trapezoidal rule still works because dx is calculated between each pair of x points individually, as in step 2.

Visual Quiz - 3 Questions

Look at the execution_table at step 3, what is the average y value between y[1]=1 and y[2]=4?

A 1.5

B 3

C 2.5

D 4

💡 Hint

Check the avg_y calculation in step 3 of the execution_table.

At which step do we calculate the width of each trapezoid (dx)?

A Step 2

B Step 4

C Step 1

D Step 5

💡 Hint

Look for where differences between x points are computed in the execution_table.

If the x array was [0, 2, 4, 6], how would the dx array change?

A [1,1,1]

B [2,2,2]

C [0,2,4]

D [6,6,6]

💡 Hint

dx is the difference between consecutive x values, see step 2 in variable_tracker.

Concept Snapshot

Trapezoidal rule approximates integral by summing trapezoids.
Calculate dx as differences between x points.
Average adjacent y values for trapezoid heights.
Multiply dx by average y and sum all.
Use scipy.integrate.trapezoid(y, x) for easy calculation.

Full Transcript

The trapezoidal rule estimates the area under a curve by dividing it into trapezoids between data points. We start with arrays of x and y values. First, we find the width of each trapezoid by calculating differences between x points (dx). Next, we find the average height of each trapezoid by averaging adjacent y values. Then, we multiply each width by its average height to get the area of each trapezoid. Finally, we sum all these areas to get the approximate integral. This process is shown step-by-step in the execution table. The scipy function trapezoid(y, x) automates this calculation. Key points include understanding why we average y values and multiply by dx. The method works even if x points are not evenly spaced.