Let [x] be the greatest integer function, where x is a real number, then $\int_0^1 \int_0^1 \int_0^1 ([x]+[y]+[z]) dx dy dz =$
To solve the integral $\int_0^1 \int_0^1 \int_0^1 ([x]+[y]+[z]) \, dx \, dy \, dz$, where $[x]$, $[y]$, and $[z]$ denote the greatest integer functions for $x$, $y$, and $z$ respectively, we first need to understand how the greatest integer function behaves over the interval from 0 to 1.
The greatest integer function, denoted by $[a]$, gives the greatest integer less than or equal to $a$. Hence, for any $x \in [0, 1)$, $[x] = 0$. This is similarly true for $y$ and $z$ as they are also integrated over the interval [0, 1).
Therefore, within the integration limits from 0 to 1, $[x] = 0$, $[y] = 0$, and $[z] = 0$ for all points in that range. Consequently, the expression inside the integral $( [x] + [y] + [z] )$ simplifies to 0:
$$\int_0^1 \int_0_1 \int_0^1 ([x] + [y] + [z]) \, dx \, dy \, dz = \int_0^1 \int_0_1 \int_0^1 (0) \, dx \, dy \, dz = 0.$$
Since the integrand is zero throughout the entire volume of integration, the value of the integral is indeed 0.
Thus, the correct answer is 0.
Important Questions from Mixed Topic (CUET PG)
- Who was the founder of Bolshevik Communist party?
- What is the key guide to statecraft in the realist tradition?
- Chronologically arrange the events in the Cold War period.
A. Berlin Wall is constructed
B. Communist China joins the UN
C. Soviet invasion of Czechoslovakia
D. Berlin Blockade
Choose the correct answer from the options given below: Morgenthau's principles of political realism are:
A. Politics is rooted in permanent and unchanging human nature which is basically self centred, self-regarding and self-interested
B. Politics is an autonomous sphere of action and cannot therefore be reduced to morals
C. International Politics is an arena of conflicting self-interests
D. The ethics of international relations is situational ethics which is very different from private morality
Choose the correct answer from the options given below:Who among the following political thinkers consider the anarchical self help system to be a compelling factor for States to maximise their relative power positions?
