Question

If \([x]\) is the greatest integer function not greater than \(x\), then the value of \(\int _0^2x[x^2]\,dx\) is...

• A. \(\frac{3}{2}\)
• B. \(\frac{5}{2}\)
• C. \(3\)
• D. \(5\)

Hint:

Split [0, 2] where the greatest integer [x^2] changes value.

Answer 1

The Correct Option is C

Step 1: Find the Breakpoints:
\([x^2]\) changes at \(x^2=1,2,3\), i.e. \(x=1,\sqrt2,\sqrt3\). So on \([0,1)\) it is 0, on \([1,\sqrt2)\) it is 1, on \([\sqrt2,\sqrt3)\) it is 2, and on \([\sqrt3,2)\) it is 3.

Step 2: Split the Integral:
\[ I=0+\int_1^{\sqrt2}x\,dx+2\int_{\sqrt2}^{\sqrt3}x\,dx+3\int_{\sqrt3}^{2}x\,dx \]

Step 3: Evaluate:
\(\int x\,dx=\dfrac{x^2}2\). So
\[ I=\frac{2-1}{2}+2\cdot\frac{3-2}{2}+3\cdot\frac{4-3}{2}=\frac12+1+\frac32=3 \]

Step 4: Check the Options:
(A) \(\tfrac32\) and (B) \(\tfrac52\) miss one of the three pieces, and (D) 5 overcounts. The sum of the pieces is 3.

Final Answer:
The value is 3, option (C). \[ \boxed{\text{(C) } 3} \]