Question

Let [x] be the greatest integer function, where x is a real number, then \( \int_{0}^{1} \int_{0}^{1} \int_{0}^{1} ([x] + [y] + [z]) \, dx \, dy \, dz = \)

• A. 0
• B. \(\frac{1}{3}\)
• C. 1
• D. 3

Hint:

When dealing with integrals involving step functions like the greatest integer function, always break down the domain of integration into intervals where the function is constant. In this case, the entire domain (except for boundaries of measure zero) corresponds to a single constant value for the integrand.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
This problem requires evaluating a triple integral of a function involving the greatest integer function (or floor function), denoted by \([x]\). The greatest integer function \([x]\) gives the largest integer less than or equal to \(x\).

Step 2: Key Formula or Approach:
The key to solving this problem is to analyze the value of the integrand, \([x] + [y] + [z]\), over the given domain of integration. The domain is a unit cube defined by: \( 0 \le x \le 1 \)
\( 0 \le y \le 1 \)
\( 0 \le z \le 1 \)

Step 3: Detailed Explanation:
Let's consider the values of \([x]\), \([y]\), and \([z]\) within the specified limits of integration.
For the variable \(x\), the integration is from 0 to 1. In the interval \(0 \le x<1\), the value of the greatest integer function \([x]\) is 0. At the single point \(x=1\), \([x]=1\). However, the value of an integral is not affected by the value of the function at a single point. So, for the purpose of integration over the interval \([0,1]\), we can consider \([x]=0\).
Similarly, for the variable \(y\) in the interval \(0 \le y<1\), \([y] = 0\).
And for the variable \(z\) in the interval \(0 \le z<1\), \([z] = 0\).
Therefore, inside the entire region of integration (the unit cube, excluding the faces at x=1, y=1, z=1 which have zero volume), the integrand is: \[ [x] + [y] + [z] = 0 + 0 + 0 = 0 \] The integral of a function that is zero everywhere in the domain of integration is zero. \[ \int_{0}^{1} \int_{0}^{1} \int_{0}^{1} (0) \, dx \, dy \, dz = 0 \]
Step 4: Final Answer:
The value of the triple integral is 0.