Question

The value of \(\int_0^2\left[x^2\right] \mathrm{d} x\) is (where \([x]\) denotes the greatest integer function not greater than \(x\) )

• A. \(5-\sqrt{2}-\sqrt{3}\)
• B. \(5+\sqrt{2}-\sqrt{3}\)
• C. \(5+\sqrt{2}+\sqrt{3}\)
• D. \(5-\sqrt{2}+\sqrt{3}\)

Hint:

Split the interval where the integer part of x squared stays constant, then add the pieces.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept
The value of \([x^{2}]\) is constant between points where \(x^{2}\) is an integer.
For \(0 \le x \le 2\), \(x^{2}\) goes from 0 to 4, so the breakpoints are \(1,\sqrt{2},\sqrt{3},2\).

Step 2: Values on Each Interval
On \([0,1)\): \([x^{2}]=0\).
On \([1,\sqrt{2})\): 1. On \([\sqrt{2},\sqrt{3})\): 2. On \([\sqrt{3},2)\): 3.

Step 3: Calculation
\(\int_0^2[x^{2}]dx=1(\sqrt{2}-1)+2(\sqrt{3}-\sqrt{2})+3(2-\sqrt{3})\).
\(=\sqrt{2}-1+2\sqrt{3}-2\sqrt{2}+6-3\sqrt{3}\).

Step 4: Simplify
\(=5-\sqrt{2}-\sqrt{3}\), which is option (A).

Final Answer:
The integral equals \(5-\sqrt{2}-\sqrt{3}\). \[ \boxed{\text{(A) }5-\sqrt{2}-\sqrt{3}} \]