Triple integral (tplquad) in SciPy - Step-by-Step Execution

Concept Flow - Triple integral (tplquad)

Define integrand function f(x,y,z)

↓

Set integration limits for z (inner integral)

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Set integration limits for y (middle integral)

↓

Set integration limits for x (outer integral)

↓

Call scipy.integrate.tplquad

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Compute inner integral over z

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Compute middle integral over y

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Compute outer integral over x

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Return total integral value and error estimate

The triple integral is computed by integrating the function first over z, then y, then x, using nested limits and scipy's tplquad function.

Execution Sample

SciPy

from scipy.integrate import tplquad

f = lambda z, y, x: x*y*z

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

This code calculates the triple integral of f(x,y,z) = x*y*z over the cube [0,1]x[0,1]x[0,1].

Execution Table

Step	Action	Variables	Evaluation	Result
1	Define integrand f(z,y,x)	f	f(z,y,x) = x*y*z	Function ready
2	Set outer limits for x	x	0 to 1	Limits set
3	Set middle limits for y	y	0 to 1	Limits set
4	Set inner limits for z	z	0 to 1	Limits set
5	Integrate over z for fixed x,y	Integral_z	∫0^1 x*y*z dz	x*y*0.5
6	Integrate over y for fixed x	Integral_y	∫0^1 x*y*0.5 dy	x*0.25
7	Integrate over x	Integral_x	∫0^1 x*0.25 dx	0.125
8	Return result and error	result, error	0.125, small error	Integral = 0.125

💡 Integration complete over all variables within limits

Variable Tracker

Variable	Start	After Step 5	After Step 6	After Step 7	Final
Integral_z	None	x*y*0.5	x*y*0.5	x*y*0.5	x*y*0.5
Integral_y	None	None	x*0.25	x*0.25	x*0.25
Integral_x	None	None	None	0.125	0.125
result	None	None	None	None	0.125

Key Moments - 3 Insights

Why does the integrand function f have arguments in the order (z, y, x) instead of (x, y, z)?

Because scipy's tplquad expects the integrand with the innermost variable first (z), then middle (y), then outer (x). This matches the integration order shown in the execution_table rows 5-7.

How do the lambda functions for limits work for y and z depending on x and y?

The limits for y and z can depend on outer variables. Here they are constant (0 to 1), but the lambdas allow dynamic limits. This is shown in steps 3 and 4 of the execution_table.

Why is the integration done in the order z, then y, then x?

Because tplquad integrates from the innermost variable to the outermost. This order is reflected in the stepwise integration in the execution_table rows 5-7.

Visual Quiz - 3 Questions

Look at the execution_table at step 5, what is the result of integrating over z for fixed x and y?

A 0.5

B x * y

C x * y * 0.5

D x * y * 1

💡 Hint

Check the 'Result' column in execution_table row 5.

At which step does the integration over y happen according to the execution_table?

A Step 5

B Step 6

C Step 7

D Step 8

💡 Hint

Look at the 'Action' column for integration over y in the execution_table.

If the limit for z changed from 0 to 2, how would the integral over z at step 5 change?

A It would become x * y * 2

B It would become x * y * 0.5

C It would become x * y * 1

D It would not change

💡 Hint

Recall integral of z from 0 to a is (a^2)/2, so doubling upper limit doubles the integral result.

Concept Snapshot

Triple integral with tplquad:
- Integrand f(z,y,x) with innermost variable first
- Limits: outer x, middle y, inner z
- Call: tplquad(f, x0, x1, y_low, y_high, z_low, z_high)
- Integrates inner to outer: z, y, then x
- Returns integral value and error estimate

Full Transcript

This visual execution traces how scipy's tplquad computes a triple integral. The integrand function is defined with variables in the order (z,y,x) because tplquad integrates from innermost to outermost variable. The integration limits are set for x, y, and z, with y and z limits possibly depending on outer variables. The integration proceeds stepwise: first over z, then y, then x. Each step simplifies the integral until the final value is computed. The variable tracker shows how partial integrals evolve after each step. Key moments clarify common confusions about argument order and limit functions. The visual quiz tests understanding of integration steps and effects of changing limits. The concept snapshot summarizes the syntax and behavior of tplquad for triple integrals.

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