integral3 Vectorized input

If you are asking about computing a triple integral of a function with vectorized input, it typically involves using the vectorization capabilities of the software or programming language you are using.

For example, in Python with NumPy, you can define a function that takes a NumPy array as input and returns a NumPy array of the same shape. You can then use the NumPy function `numpy.vectorize()` to create a vectorized version of the function, which can be used to compute the triple integral for a vectorized input.

Here's an example code snippet:

import numpy as np

# Define the function to integrate
def f(x, y, z):
    return x**2 + y**2 + z**2

# Vectorize the function
f_vec = np.vectorize(f)

# Define the limits of integration
x_min, x_max = 0, 1
y_min, y_max = 0, 1
z_min, z_max = 0, 1

# Create the arrays for the triple integral inputs
x = np.linspace(x_min, x_max, 100)
y = np.linspace(y_min, y_max, 100)
z = np.linspace(z_min, z_max, 100)
dx = x[1] - x[0]
dy = y[1] - y[0]
dz = z[1] - z[0]
X, Y, Z = np.meshgrid(x, y, z, indexing='ij')

# Compute the triple integral using the trapezoidal rule
integral = dx * dy * dz * np.sum(f_vec(X, Y, Z))

This code computes the triple integral of the function `f(x, y, z) = x^2 + y^2 + z^2` over the unit cube in three dimensions, using the trapezoidal rule for numerical integration with 100 points along each axis. The input to the function `f` is vectorized using `numpy.vectorize()`, and the integral is computed by summing the vectorized function over the input arrays.