Step 1: Understanding the Concept:
The Greatest Integer Function, denoted by \([x]\), returns the largest integer less than or equal to \(x\).
The function \([x^2]\) is a step function. It changes its value only at points where the internal term \(x^2\) becomes an integer.
When integrating a step function over an interval \([a, b]\), we must split the integral into sub-intervals based on these "critical points" where the value jumps.
Within each sub-interval, the function remains constant, turning the integration into a series of area calculations for rectangles.
Step 2: Key Formula or Approach:
We identify the critical points for \(x^2\) in the interval \(x \in [1, 2]\).
As \(x\) goes from \(1\) to \(2\), \(x^2\) goes from \(1^2 = 1\) to \(2^2 = 4\).
The integer values that \(x^2\) takes in this range are \(1, 2, 3,\) and \(4\).
The critical values of \(x\) are therefore:
- \(x^2 = 1 \implies x = 1\)
- \(x^2 = 2 \implies x = \sqrt{2}\)
- \(x^2 = 3 \implies x = \sqrt{3}\)
- \(x^2 = 4 \implies x = 2\)
Step 3: Detailed Explanation:
We divide the integral into three parts based on these values:
\[ I = \int_{1}^{\sqrt{2}} [x^2] dx + \int_{\sqrt{2}}^{\sqrt{3}} [x^2] dx + \int_{\sqrt{3}}^{2} [x^2] dx \]
Now, let's determine the value of \([x^2]\) in each sub-interval:
1. For \(1 \leq x<\sqrt{2}\), we have \(1 \leq x^2<2\). Thus, \([x^2] = 1\).
2. For \(\sqrt{2} \leq x<\sqrt{3}\), we have \(2 \leq x^2<3\). Thus, \([x^2] = 2\).
3. For \(\sqrt{3} \leq x<2\), we have \(3 \leq x^2<4\). Thus, \([x^2] = 3\).
Now, we perform the simple constant integration:
\[ I = \int_{1}^{\sqrt{2}} 1 dx + \int_{\sqrt{2}}^{\sqrt{3}} 2 dx + \int_{\sqrt{3}}^{2} 3 dx \]
\[ I = [x]_{1}^{\sqrt{2}} + [2x]_{\sqrt{2}}^{\sqrt{3}} + [3x]_{\sqrt{3}}^{2} \]
Evaluating the bounds:
\[ I = (\sqrt{2} - 1) + (2\sqrt{3} - 2\sqrt{2}) + (3(2) - 3\sqrt{3}) \]
\[ I = \sqrt{2} - 1 + 2\sqrt{3} - 2\sqrt{2} + 6 - 3\sqrt{3} \]
Combining the constant and radical terms:
\[ I = (6 - 1) + (\sqrt{2} - 2\sqrt{2}) + (2\sqrt{3} - 3\sqrt{3}) \]
\[ I = 5 - \sqrt{2} - \sqrt{3} \]
Step 4: Final Answer:
The value of the definite integral is \(5 - \sqrt{2} - \sqrt{3}\).