Triple integral (tplquad) in SciPy - Interactive Code Practice

Practice - 5 Tasks

1fill in blank

easy

Complete the code to import the function for triple integration from scipy.

SciPy

from scipy.integrate import [1]

Drag options to blanks, or click blank then click option'

Atplquad

Bnquad

Cdblquad

Dquad

Attempts:

3 left

💡 Hint

Common Mistakes

Using dblquad which is for double integrals.

Using quad which is for single integrals.

The tplquad function is used for triple integrals in scipy.

2fill in blank

medium

Complete the code to define the function f(x, y, z) = x * y * z for integration.

SciPy

def f(x, y, z):
    return [1]

Drag options to blanks, or click blank then click option'

Ax + y + z

Bx / (y + z)

Cx - y - z

Dx * y * z

Attempts:

3 left

💡 Hint

Common Mistakes

Adding variables instead of multiplying.

Using division which is not the function defined.

The function to integrate is the product of x, y, and z.

3fill in blank

hard

Fix the error in the integration call by completing the limits for z from 0 to 1.

SciPy

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

Drag options to blanks, or click blank then click option'

A0

Blambda x: 0

Clambda x, y: 0

D1

Attempts:

3 left

💡 Hint

Common Mistakes

Using a lambda with only one argument for z limits.

Using fixed numbers instead of functions for limits.

The limits for the innermost integral (z) must be functions of x and y, here from 0 to 1.

4fill in blank

hard

Fill both blanks to complete the integration limits for y from 0 to x and z from 0 to y.

SciPy

result, error = tplquad(f, 0, 1, [1], [2], lambda x, y: 0, lambda x, y: y)

Drag options to blanks, or click blank then click option'

Alambda x: 0

Blambda x: x

Clambda x, y: 0

Dlambda x, y: x

Attempts:

3 left

💡 Hint

Common Mistakes

Using lambdas with two arguments for y limits.

Swapping the lower and upper limits.

The limits for y are from 0 to x, so use lambda x: 0 and lambda x: x.

5fill in blank

hard

Fill all three blanks to create a dictionary comprehension that maps x to the integral of f over y and z with limits y:0 to x, z:0 to y.

SciPy

results = { [1]: tplquad(f, 0, [1], lambda x: 0, lambda x: x, [2], [3])[0] for [1] in [0.1, 0.5, 1]}

Drag options to blanks, or click blank then click option'

Ax

Blambda x, y: 0

Clambda x, y: y

Dy

Attempts:

3 left

💡 Hint

Common Mistakes

Using y as the dictionary key instead of x.

Incorrect lambda functions for z limits.

The dictionary keys are x values. The limits for z are from 0 to y, so use lambda x, y: 0 and lambda x, y: y.

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