Let [] be the greatest integer function.

Question

Let $[\cdot]$ be the greatest integer function. If $\alpha = \displaystyle\int_0^{64}\left(x^{1/3}-[x^{1/3}]\right)dx$, then $\alpha$ is equal to \_\_\_\_.

Answer: 36

Accepted Answer

$$\int_0^{64} x^{1/3}\,dx = \frac34\left[x^{4/3}\right]_0^{64} = \frac34\cdot 64^{4/3} = \frac34\cdot 256 = 192$$

For $\int_0^{64}[x^{1/3}]\,dx$, note $[x^{1/3}]=n$ for $x\in[n^3,(n+1)^3)$:

- $n=1$: $x\in[1,8)$, length $7$, contributes $7$
- $n=2$: $x\in[8,27)$, length $19$, contributes $38$
- $n=3$: $x\in[27,64)$, length $37$, contributes $111$

Total $= 7+38+111 = 156$.

So $\alpha = 192 - 156 = 36$.

Solutions · 1