Simpson's rule (simpson) in SciPy - Step-by-Step Execution

Concept Flow - Simpson's rule (simpson)

Start with data points

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Divide interval into even segments

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Apply Simpson's formula: weighted sum

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Calculate integral approximation

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Output result

Simpson's rule approximates the area under a curve by dividing it into even segments and applying a weighted sum formula to estimate the integral.

Execution Sample

SciPy

from scipy.integrate import simpson
x = [0, 1, 2, 3, 4]
y = [1, 2, 0, 2, 1]
result = simpson(y, x)
print(result)

This code calculates the integral approximation of y with respect to x using Simpson's rule.

Execution Table

Step	Action	x values	y values	Weights applied	Weighted sum	Result
1	Input data points	[0,1,2,3,4]	[1,2,0,2,1]	N/A	N/A	N/A
2	Check segments count	5 points -> 4 segments (even)	N/A	N/A	N/A	Proceed
3	Apply Simpson's weights	N/A	N/A	[1,4,2,4,1]	Calculate sum: 1*1 + 4*2 + 2*0 + 4*2 + 1*1 = 1+8+0+8+1=18	N/A
4	Calculate integral	N/A	N/A	N/A	Integral ≈ (h/3)*sum = (1/3)*18 = 6.0	6.0
5	Output result	N/A	N/A	N/A	N/A	6.0

💡 All steps completed, integral approximation calculated.

Variable Tracker

Variable	Start	After Step 2	After Step 3	After Step 4	Final
x	[0,1,2,3,4]	[0,1,2,3,4]	[0,1,2,3,4]	[0,1,2,3,4]	[0,1,2,3,4]
y	[1,2,0,2,1]	[1,2,0,2,1]	[1,2,0,2,1]	[1,2,0,2,1]	[1,2,0,2,1]
weights	N/A	N/A	[1,4,2,4,1]	[1,4,2,4,1]	[1,4,2,4,1]
weighted_sum	N/A	N/A	18	18	18
result	N/A	N/A	N/A	6.0	6.0

Key Moments - 3 Insights

Why must the number of segments be even for Simpson's rule?

Simpson's rule requires pairs of intervals to apply the quadratic approximation. The execution_table step 2 shows 4 segments (even), allowing the weighted sum in step 3.

How are the weights [1,4,2,4,1] determined?

Weights alternate between 4 and 2 for interior points, with 1 at the ends. This pattern is shown in execution_table step 3 where weights multiply y values.

What does the 'h' value represent in the formula?

'h' is the width of each segment. Here, x values are equally spaced by 1, so h=1. This is used in step 4 to calculate the integral as (h/3)*sum.

Visual Quiz - 3 Questions

Look at the execution_table step 3, what is the weighted sum calculated?

A 18

B 12

C 24

D 6

💡 Hint

Check the 'Weighted sum' column in step 3 of execution_table.

At which step does the code confirm the number of segments is even?

A Step 1

B Step 3

C Step 2

D Step 4

💡 Hint

Look at the 'Action' column describing segment count in execution_table.

If the x values were spaced by 0.5 instead of 1, how would the final result change?

A It would double

B It would halve

C It would stay the same

D It would quadruple

💡 Hint

Recall in variable_tracker 'result' depends on h, which is segment width used in step 4.

Concept Snapshot

Simpson's rule approximates integrals using weighted sums.
Divide interval into even segments.
Weights: 1 at ends, 4 and 2 alternating inside.
Integral ≈ (h/3) * weighted sum of y values.
Use scipy.integrate.simpson(y, x) for easy calculation.

Full Transcript

Simpson's rule estimates the area under a curve by dividing the x-axis into an even number of segments. Each y value is multiplied by a weight: 1 for the first and last points, 4 and 2 alternating for the points in between. The weighted sum is then multiplied by one-third of the segment width to approximate the integral. The example code uses scipy's simpson function to perform this calculation on sample data points. The execution table shows each step: input data, checking segment count, applying weights, calculating the weighted sum, and finally computing the integral result.