Question

If \([x]\) denotes the greatest integer less than or equal to \(x\), then the value of the integral \(\int _0^2x^2[x]\,dx\) is equal to

• A. \(\frac{8}{3}\)
• B. \(\frac{3}{8}\)
• C. \(\frac{7}{3}\)
• D. \(0\)

Hint:

Split the interval at 1: the floor is 0 on [0,1) and 1 on [1,2).

Answer 1

The Correct Option is C

Step 1: Break at the jump:
The integrand changes form at $x = 1$. On $[0,1)$ it is 0, so that piece adds nothing.

Step 2: The remaining piece:
From 1 to 2, $[x] = 1$ and the integrand is $x^2$. Its area is $\frac{2^3 - 1^3}{3}$.

Step 3: Arithmetic:
$\frac{8-1}{3} = \frac73$.

Final Answer:
The value is 7/3, option (C). \[ \boxed{\frac{7}{3}} \]