Triple integral (tplquad) in SciPy - Cheat Sheet & Quick Revision

Recall & Review

beginner

What is the purpose of the tplquad function in SciPy?

tplquad is used to calculate the value of a triple integral over a three-dimensional region. It helps find the volume under a surface in 3D space.

beginner

What are the main inputs required by tplquad?

You need to provide:

• A function of three variables (z, y, x) to integrate.
• Limits for the outer integral (usually x).
• Functions or constants for the inner limits (y and z), which can depend on outer variables.

intermediate

How does tplquad handle variable limits for inner integrals?

The limits for y and z can be functions that depend on the outer variables. This allows integration over complex shapes where the bounds change with position.

beginner

What does the output of tplquad include?

It returns a tuple: the first value is the integral result (a number), and the second is an estimate of the absolute error in the result.

beginner

Write a simple example of using tplquad to integrate f(x,y,z) = x + y + z over the cube where x, y, z go from 0 to 1.

<pre>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(f"Integral result: {result}")</pre>

What does the tplquad function compute?

A A derivative of a function

B A double integral over a 2D region

C A triple integral over a 3D region

D A single integral over a 1D interval

tplquad is specifically designed for triple integrals in three dimensions.

In tplquad, what order are the variables integrated in the function signature f(z, y, x)?

A z, y, x

B x, y, z

C y, z, x

D x, z, y

The function f must take variables in the order z, y, x because integration is done inside-out.

Which of these can be used as limits for inner integrals in tplquad?

A Only functions of the outer variables

B Functions or constants

C Only constants

D Only variables

Limits can be constants or functions depending on outer variables to define complex regions.

What does the second value returned by tplquad represent?

A The integral value

B The time taken to compute

C The number of iterations

D The estimated error of the integral

The second value is an estimate of the absolute error in the integral calculation.

If you want to integrate over a cube from 0 to 1 in all directions, what are the limits for tplquad?

A All limits are 0 to 1 constants

B Outer limits 0 to 1, inner limits 0 to x

C Outer limits 0 to 1, inner limits 0 to y

D Outer limits 0 to 1, inner limits 1 to 2

For a cube from 0 to 1 in x, y, z, all limits are constants 0 to 1.

Explain how to set up a triple integral using tplquad for a function f(x,y,z) over a region where the limits of y and z depend on x.

Remember the order of integration and how limits can be functions.

Describe the output of tplquad and what each part means.

Think about what you get after integration and how reliable it is.

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