1. The electric charge that flows in a time interval is given by

$Q=\int_{t_1}^{t_2} I(t)\,dt$

Here,

$I(t)=3t^2+4t^3$

and the interval is from $t=1\,\text{s}$ to $t=2\,\text{s}$.

2. Substitute into the formula:

$Q=\int_1^2 (3t^2+4t^3)\,dt$

3. Integrate term by term:

$\int 3t^2\,dt=t^3$

$\int 4t^3\,dt=t^4$

So,

$Q=\left[t^3+t^4\right]_1^2$

4. Apply the limits:

At $t=2$,

$2^3+2^4=8+16=24$

At $t=1$,

$1^3+1^4=1+1=2$

Therefore,

$Q=24-2=22\,\text{C}$

5. Final answer:

$\boxed{22}$