Animated Solution for Mathematics - Definite Integration: The integral ${\int }_0^{1.5}[x^2]dx$, where $[]$ denotes the greatest integer function, equals $\dots \dots \dots$

The Greatest Integer Function, denoted by $[x^2]$, behaves like a series of flat, constant plateaus rather than a smooth curve. To evaluate the integral $I={\int }_0^{1.5}[x^2]dx$, we must identify the points where the function value shifts.

The function $[x^2]$ changes its value whenever $x^2$ reaches an integer. As $x$ varies from $0$ to $1.5$, the value of $x^2$ ranges from $0$ to $2.25$.

Within the interval $x^2\in [0,2.25]$, the function encounters integer values at $x^2=1$ and $x^2=2$. Solving for $x$, we identify the critical points at $x=1$ and $x=\sqrt{2}\approx 1.414$.

We can now decompose the integral into three distinct intervals based on these critical points. This allows us to treat each segment as a simple rectangle:

1. The First Interval: $x\in [0,1)$ In this range, $0\le x^2<1$, so $[x^2]=0$. The integral is:

2. The Second Interval: $x\in [1,\sqrt{2})$ In this range, $1\le x^2<2$, so $[x^2]=1$. The integral is:

3. The Final Interval: $x\in [\sqrt{2},1.5]$ In this range, $2\le x^2\le 2.25$, so $[x^2]=2$. The integral is:

To find the total value of the integral, we sum the results of the three segments:

Substituting the calculated values:

Grouping the rational and irrational terms, we arrive at the final result: $I=2-\sqrt{2}$