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

Concept Flow - Simpson's rule (simpson)

Start with data points

↓

Divide interval into even segments

↓

Apply Simpson's formula: weighted sum

↓

Calculate integral approximation

↓

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.

Practice

1. What is the main purpose of Simpson's rule in data science?

easy

A. To find the maximum value in a dataset

B. To estimate the area under a curve using parabolas

C. To sort data points in ascending order

D. To calculate the mean of a list of numbers

Solution

1. Step 1: Understand the goal of Simpson's rule

   Simpson's rule is a numerical method used to estimate the area under a curve by approximating the curve with parabolas between points.

2. Step 2: Compare options with this goal

   Only To estimate the area under a curve using parabolas describes this purpose correctly. Other options describe unrelated tasks.

3. Final Answer:

   To estimate the area under a curve using parabolas -> Option B

4. Quick Check:

   Simpson's rule = area estimation [OK]

Hint: Simpson's rule estimates area under curves using parabolas [OK]

Common Mistakes:

• Confusing Simpson's rule with sorting or averaging
• Thinking it finds max or min values
• Assuming it requires a formula for the curve

2. Which of the following is the correct way to import the Simpson's rule function from scipy for numerical integration?

easy

A. from scipy.integrate import simpson

B. import scipy.simpson

C. from scipy import simpson

D. import simpson from scipy.integrate

Solution

1. Step 1: Recall the correct import syntax for functions in scipy

   Functions like simpson are in the integrate module, so the correct import is from scipy.integrate import simpson.

2. Step 2: Check each option

   from scipy.integrate import simpson matches the correct syntax. Options A, B, and D are incorrect. import simpson from scipy.integrate uses wrong import order.

3. Final Answer:

   from scipy.integrate import simpson -> Option A

4. Quick Check:

   Correct import = from scipy.integrate import simpson [OK]

Hint: Import simpson from scipy.integrate module [OK]

Common Mistakes:

• Trying to import simpson directly from scipy
• Using wrong import syntax order
• Assuming simpson is a standalone package

3. What is the output of the following code?

from scipy.integrate import simpson
import numpy as np
x = np.array([0, 1, 2, 3, 4])
y = np.array([0, 1, 4, 9, 16])
result = simpson(y, x)
print(round(result, 2))

medium

A. 22.67

B. 18.00

C. 20.00

D. 21.33

Solution

1. Step 1: Understand the data points and function

   x values are [0,1,2,3,4], y values are squares of x: [0,1,4,9,16]. Simpson's rule estimates the integral of y with respect to x.

2. Step 2: Calculate the integral using simpson

   Using scipy's simpson function on these points gives exactly 21.3333. Rounded to two decimals, it is 21.33.

3. Final Answer:

   21.33 -> Option D

4. Quick Check:

   Simpson integral ≈ 21.33 [OK]

Hint: Use simpson(y, x) and round result for quick check [OK]

Common Mistakes:

• Forgetting to provide x values to simpson
• Rounding incorrectly or too early
• Confusing simpson with trapezoidal rule

4. Identify the error in this code snippet using Simpson's rule:

from scipy.integrate import simpson
import numpy as np
x = np.array([0, 1, 2, 3])
y = np.array([1, 4, 9])
result = simpson(y, x)
print(result)

medium

A. x and y arrays have different lengths

B. simpson function is not imported correctly

C. x array must be sorted in descending order

D. y array contains non-numeric values

Solution

1. Step 1: Check array lengths

   x has 4 elements, y has 3 elements. Simpson's rule requires x and y to have the same length.

2. Step 2: Verify other options

   Import is correct, x order is fine, y contains numeric values. So only length mismatch is the error.

3. Final Answer:

   x and y arrays have different lengths -> Option A

4. Quick Check:

   Length mismatch error = x and y arrays have different lengths [OK]

Hint: Check x and y array lengths match before integration [OK]

Common Mistakes:

• Ignoring length mismatch causing runtime error
• Assuming simpson auto-corrects array sizes
• Thinking x must be descending

5. You have noisy sensor data sampled unevenly over time:

time = [0, 0.5, 1.5, 2.2, 3.8]
values = [2, 3, 5, 4, 6]

How can you use Simpson's rule to estimate the total area under the sensor curve? Choose the best approach.

hard

A. Interpolate values to evenly spaced points, then use simpson without x

B. Use simpson(values) without x, ignoring time points

C. Use simpson(values, time) directly to integrate over uneven intervals

D. Sum values and multiply by average time interval

Solution

1. Step 1: Understand uneven sampling

   Time points are unevenly spaced, so integration must consider x values explicitly.

2. Step 2: Apply simpson with x values

   Passing values and time arrays to simpson allows correct numerical integration over uneven intervals.

3. Step 3: Evaluate other options

   Interpolating or ignoring x loses accuracy; summing values times average interval is a rough estimate, not Simpson's rule.

4. Final Answer:

   Use simpson(values, time) directly to integrate over uneven intervals -> Option C

5. Quick Check:

   Simpson with x for uneven data = Use simpson(values, time) directly to integrate over uneven intervals [OK]

Hint: Always provide x for unevenly spaced data in simpson [OK]

Common Mistakes:

• Ignoring uneven spacing and skipping x parameter
• Using interpolation unnecessarily
• Using simple sum times average interval instead of integration