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.