Simpson's rule (simpson) in SciPy - Mini Project: Build & Apply

Calculate Area Under Curve Using Simpson's Rule

📖 Scenario: Imagine you are a scientist measuring the speed of a car at different times. You want to find out how far the car traveled during that time by calculating the area under the speed-time graph.

🎯 Goal: You will use Simpson's rule to calculate the approximate area under the curve from given speed measurements at specific times.

📋 What You'll Learn

Create a list called time_points with exact values: 0, 1, 2, 3, 4

Create a list called speed_measurements with exact values: 0, 10, 20, 10, 0

Create a variable called dx and set it to 1

Use scipy.integrate.simpson to calculate the area under the curve using speed_measurements and dx

Print the calculated area with the exact variable name area

💡 Why This Matters

🌍 Real World

Simpson's rule helps scientists and engineers estimate areas under curves when exact formulas are hard to find, such as calculating distance from speed data.

💼 Career

Understanding numerical integration like Simpson's rule is useful in data analysis, physics, engineering, and any job involving measurement and estimation from data.

1

Create time and speed data lists

Create a list called time_points with values 0, 1, 2, 3, 4 and a list called speed_measurements with values 0, 10, 20, 10, 0.

SciPy

# Create the lists time_points and speed_measurements
# Your code here

Hint

Use square brackets to create lists with the exact numbers given.

2

Set the spacing variable dx

Create a variable called dx and set it to 1 to represent the equal spacing between time points.

SciPy

time_points = [0, 1, 2, 3, 4]
speed_measurements = [0, 10, 20, 10, 0]
# Create the variable dx and set it to 1
# Your code here

Hint

Just assign the number 1 to the variable dx.

3

Calculate area using Simpson's rule

Import simpson from scipy.integrate and use it to calculate the area under the curve by passing speed_measurements and dx. Store the result in a variable called area.

SciPy

time_points = [0, 1, 2, 3, 4]
speed_measurements = [0, 10, 20, 10, 0]
dx = 1
# Import simpson and calculate area using simpson(speed_measurements, dx=dx)
# Store the result in area
# Your code here

Hint

Use from scipy.integrate import simpson and then call simpson(speed_measurements, dx=dx).

4

Print the calculated area

Write a print statement to display the value of the variable area.

SciPy

from scipy.integrate import simpson

time_points = [0, 1, 2, 3, 4]
speed_measurements = [0, 10, 20, 10, 0]
dx = 1

area = simpson(speed_measurements, dx=dx)
# Print the area
# Your code here

Hint

Use print(area) to show the 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