Triple integral (tplquad) in SciPy - Mini Project: Build & Apply

Calculate Volume Using Triple Integral with tplquad

📖 Scenario: Imagine you are a scientist studying a 3D shape defined by a mathematical function. You want to find the volume under this shape within certain limits.

🎯 Goal: You will calculate the volume under the function f(x, y, z) = x + y + z over a specific 3D region using the tplquad function from scipy.integrate.

📋 What You'll Learn

Create a function f that takes three variables x, y, and z and returns x + y + z.

Set the integration limits for x from 0 to 1.

Set the integration limits for y from 0 to 2.

Set the integration limits for z from 0 to 3.

Use tplquad to calculate the triple integral of f over these limits.

Print the calculated volume.

💡 Why This Matters

🌍 Real World

Triple integrals are used in physics and engineering to calculate volumes, masses, and other quantities in three-dimensional spaces.

💼 Career

Understanding how to compute triple integrals helps in fields like data science, physics simulations, and engineering design where 3D data and volumes are analyzed.

1

Define the function to integrate

Create a function called f that takes three arguments x, y, and z and returns the sum x + y + z.

SciPy

# Define the function f(x, y, z) that returns x + y + z
# Your code here

Hint

Remember to define a function using def and return the sum of the three inputs.

2

Set the integration limits

Create three variables: x_lower set to 0, x_upper set to 1, y_lower set to 0, y_upper set to 2, z_lower set to 0, and z_upper set to 3.

SciPy

def f(x, y, z):
    return x + y + z

# Set integration limits for x, y, and z
# Your code here

Hint

Assign the exact numbers to the variables as given.

3

Calculate the triple integral using tplquad

Import tplquad from scipy.integrate. Then use tplquad to calculate the triple integral of f over x from x_lower to x_upper, y from y_lower to y_upper, and z from z_lower to z_upper. Store the result in a variable called volume.

SciPy

def f(x, y, z):
    return x + y + z

x_lower = 0
x_upper = 1
y_lower = 0
y_upper = 2
z_lower = 0
z_upper = 3

# Import tplquad and calculate the triple integral
# Your code here

Hint

Remember the order of integration in tplquad is func, x_lower, x_upper, y_lower_func, y_upper_func, z_lower_func, z_upper_func. Use lambda functions for the y and z limits.

4

Print the calculated volume

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

SciPy

from scipy.integrate import tplquad

def f(x, y, z):
    return x + y + z

x_lower = 0
x_upper = 1
y_lower = 0
y_upper = 2
z_lower = 0
z_upper = 3

volume, error = tplquad(f, x_lower, x_upper, lambda x: y_lower, lambda x: y_upper, lambda x, y: z_lower, lambda x, y: z_upper)

# Print the volume
# Your code here

Hint

Use print(volume) to show the result.

Practice

1.

What does the tplquad function in SciPy compute?

easy

A. The derivative of a function with three variables

B. The value of a triple integral over a 3D region

C. The sum of three separate integrals

D. The product of three functions

Solution

1. Step 1: Understand the purpose of tplquad

   The tplquad function is designed to calculate triple integrals, which means integrating a function over a three-dimensional space.

2. Step 2: Differentiate from other operations

   It does not compute derivatives, sums of integrals, or products of functions. Its main role is to compute the total accumulation (integral) over a volume.

3. Final Answer:

   The value of a triple integral over a 3D region -> Option B

4. Quick Check:

   tplquad = triple integral calculation [OK]

Hint: Remember tplquad integrates over three variables [OK]

Common Mistakes:

• Confusing integration with differentiation
• Thinking tplquad sums separate integrals
• Assuming tplquad multiplies functions

2.

Which of the following is the correct syntax to call tplquad for integrating f(x, y, z) over x from 0 to 1, y from 0 to 2, and z from 0 to 3?

from scipy.integrate import tplquad
result = tplquad(____, 0, 1, lambda x: 0, lambda x: 2, lambda x, y: 0, lambda x, y: 3)

easy

A. f, 0, 1, 0, 2, 0, 3

B. f, 0, 1, lambda x: 0, 2, lambda x, y: 0, 3

C. f, 0, 1, lambda x: 0, lambda x: 2, lambda x, y: 0, lambda x, y: 3

D. f, 0, 1, 0, 2, lambda x: 0, lambda x: 3

Solution

1. Step 1: Understand tplquad parameter order

   The function signature is tplquad(func, x_min, x_max, y_min_func, y_max_func, z_min_func, z_max_func). The limits for y and z must be functions of the outer variables.

2. Step 2: Check the given options

   Only f, 0, 1, lambda x: 0, lambda x: 2, lambda x, y: 0, lambda x, y: 3 correctly uses lambda functions for y and z limits, matching the required syntax.

3. Final Answer:

   f, 0, 1, lambda x: 0, lambda x: 2, lambda x, y: 0, lambda x, y: 3 -> Option C

4. Quick Check:

   Limits as lambdas for inner variables = correct syntax [OK]

Hint: Use lambdas for inner integral limits in tplquad [OK]

Common Mistakes:

• Using numbers directly instead of functions for inner limits
• Mixing order of limits
• Omitting lambda for inner limits

3.

What is the output of this code?

from scipy.integrate import tplquad

def f(z, y, x):
    return x + y + z

result, error = tplquad(f, 0, 1, lambda x: 0, lambda x: 1, lambda x, y: 0, lambda x, y: 1)
print(round(result, 2))

medium

A. 1.5

B. 1.0

C. 3.0

D. 0.5

Solution

1. Step 1: Understand the function and integration limits

   The function is f(z,y,x) = x + y + z. The integration limits for x, y, z are all from 0 to 1.

2. Step 2: Calculate the triple integral

   Integral over x: ∫0^1 x dx = 0.5
   Integral over y: ∫0^1 y dy = 0.5
   Integral over z: ∫0^1 z dz = 0.5
   Total integral = ∫∫∫ (x + y + z) dz dy dx = sum of integrals over each variable = 3 * 0.5 = 1.5

3. Final Answer:

   1.5 -> Option A

4. Quick Check:

   Sum of integrals = 1.5 [OK]

Hint: Sum integrals of x, y, z separately over 0 to 1 [OK]

Common Mistakes:

• Confusing order of variables in function
• Forgetting to integrate over all three variables
• Incorrectly summing integrals

4.

Identify the error in this code snippet using tplquad:

from scipy.integrate import tplquad

def f(x, y, z):
    return x * y * z

result = tplquad(f, 0, 1, 0, 1, 0, 1)
print(result[0])

medium

A. Integration limits for y and z should be functions, not numbers

B. Missing import statement for tplquad

C. Function parameters order is incorrect for tplquad

D. Print statement syntax is wrong

Solution

1. Step 1: Check function parameter order

   In tplquad, the function should have parameters in order (z, y, x). Here it is (x, y, z), which is incorrect.

2. Step 2: Check integration limits

   Limits for y and z must be functions of outer variables, not constants. Passing numbers directly causes an error.

3. Final Answer:

   Integration limits for y and z should be functions, not numbers -> Option A

4. Quick Check:

   Inner limits must be functions [OK]

Hint: Use lambdas for inner limits, function args order is (z,y,x) [OK]

Common Mistakes:

• Passing constants instead of functions for inner limits
• Wrong order of function parameters
• Assuming tplquad accepts numeric limits for all variables

5.

You want to calculate the volume under the surface f(x, y, z) = x * y * z where 0 <= x <= 2, 0 <= y <= x, and 0 <= z <= y. Which tplquad call correctly computes this volume?

hard

A. tplquad(lambda z, y, x: x*y*z, 0, 2, lambda x: 0, lambda x: y, lambda x, y: 0, lambda x, y: y)

B. tplquad(lambda x, y, z: x*y*z, 0, 2, lambda x: 0, lambda x: x, lambda x, y: 0, lambda x, y: y)

C. tplquad(lambda z, y, x: x*y*z, 0, 2, 0, lambda x: x, 0, lambda x, y: y)

D. tplquad(lambda z, y, x: x*y*z, 0, 2, lambda x: 0, lambda x: x, lambda x, y: 0, lambda x, y: y)

Solution

1. Step 1: Confirm function parameter order

   In tplquad, the function must have parameters in order (z, y, x). tplquad(lambda z, y, x: x*y*z, 0, 2, lambda x: 0, lambda x: x, lambda x, y: 0, lambda x, y: y) uses this order correctly.

2. Step 2: Check integration limits match the problem

   x goes from 0 to 2.
   y goes from 0 to x (lambda x: 0 to lambda x: x).
   z goes from 0 to y (lambda x, y: 0 to lambda x, y: y). tplquad(lambda z, y, x: x*y*z, 0, 2, lambda x: 0, lambda x: x, lambda x, y: 0, lambda x, y: y) matches these limits correctly.

3. Final Answer:

   tplquad(lambda z, y, x: x*y*z, 0, 2, lambda x: 0, lambda x: x, lambda x, y: 0, lambda x, y: y) -> Option D

4. Quick Check:

   Function args (z,y,x) and limits match problem [OK]

Hint: Match function args (z,y,x) and use lambdas for limits [OK]

Common Mistakes:

• Wrong function argument order
• Using constants instead of lambdas for limits
• Mixing up limit functions for y and z