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: Layer Idea:
Write the integral as a sum of strips: $I=\sum k\int_{\sqrt{k}}^{\sqrt{k+1}}x\,dx$ over $k=0,1,2,3$ (stopping at 2).

Step 2: Each Strip:
Each strip gives $k\cdot\dfrac{(k+1)-k}{2}=\dfrac k2$.

Step 3: Sum:
$\dfrac{0+1+2+3}{2}=3$. Option (C).

Final Answer:
Option (C). \[ \boxed{\text{(C) } 3} \]