1. We need to evaluate $I=\int_0^3\left(\lfloor x^2\rfloor+\left\lfloor \frac{x^2}{2}\right\rfloor\right)dx.$

We will compute the two integrals separately: $I_1=\int_0^3 \lfloor x^2\rfloor \,dx, \qquad I_2=\int_0^3 \left\lfloor \frac{x^2}{2}\right\rfloor dx.$ Then $I=I_1+I_2.$

2. Compute $I_1=\int_0^3 \lfloor x^2\rfloor dx$

Since $x\in[0,3]$, we have $x^2\in[0,9]$. The value of $\lfloor x^2\rfloor=n$ when $n\le x^2 < n+1 \quad \Rightarrow \quad \sqrt n \le x < \sqrt{n+1}.$ So: $I_1=\sum_{n=0}^{8} n\,\bigl(\sqrt{n+1}-\sqrt n\bigr).$ Write termwise: \begin{align*} I_1&=0(1-0)+1(\sqrt2-1)+2(\sqrt3-\sqrt2)+3(2-\sqrt3)\ &\quad+4(\sqrt5-2)+5(\sqrt6-\sqrt5)+6(\sqrt7-\sqrt6)+7(\sqrt8-\sqrt7)+8(3-\sqrt8). \end{align*} Now combine like terms.

Constants: $-1+6-8+24=21.$

Radicals:

• $\sqrt2$: $1-2=-1$
• $\sqrt3$: $2-3=-1$
• $\sqrt5$: $4-5=-1$
• $\sqrt6$: $5-6=-1$
• $\sqrt7$: $6-7=-1$
• $\sqrt8$: $7-8=-1$

Hence $I_1=21-\sqrt2-\sqrt3-\sqrt5-\sqrt6-\sqrt7-\sqrt8.$ Since $\sqrt8=2\sqrt2$, $I_1=21-3\sqrt2-\sqrt3-\sqrt5-\sqrt6-\sqrt7.$

3. Compute $I_2=\int_0^3 \left\lfloor \frac{x^2}{2}\right\rfloor dx$

Now $\dfrac{x^2}{2}\in\left[0,\dfrac92\right]$, so possible integer values are $0,1,2,3,4$.

We have $k\le \frac{x^2}{2}<k+1 \quad \Rightarrow \quad \sqrt{2k}\le x<\sqrt{2k+2}.$ Therefore $I_2=\sum_{k=0}^{3} k\,\bigl(\sqrt{2k+2}-\sqrt{2k}\bigr)+4\,(3-\sqrt8).$ That is, \begin{align*} I_2&=0(\sqrt2-0)+1(2-\sqrt2)+2(\sqrt6-2)+3(\sqrt8-\sqrt6)+4(3-\sqrt8). \end{align*} Now simplify.

Constants: $2-4+12=10.$

Radicals:

• $\sqrt2$: $-\sqrt2$
• $\sqrt6$: $2\sqrt6-3\sqrt6=-\sqrt6$
• $\sqrt8$: $3\sqrt8-4\sqrt8=-\sqrt8=-2\sqrt2$

Thus $I_2=10-3\sqrt2-\sqrt6.$

4. Add the two parts \begin{align*} I&=I_1+I_2\ &=\left(21-3\sqrt2-\sqrt3-\sqrt5-\sqrt6-\sqrt7\right)+\left(10-3\sqrt2-\sqrt6\right)\ &=31-6\sqrt2-\sqrt3-\sqrt5-2\sqrt6-\sqrt7. \end{align*}

This matches the form $a+b\sqrt2-\sqrt3-\sqrt5+c\sqrt6-\sqrt7,$ so we get $a=31,\quad b=-6,\quad c=-2.$ Therefore $a+b+c=31-6-2=23.$

5. Comparison with stored answer

Derived answer: $23$. Stored correct answer: $23$. So they agree.