The rate at which water is sprayed on a field of vegetables is given by $R(t)=3\sqrt{1+3t^3}$, where $t$ is in minutes and $R(t)$ is in gallons per minute. During the time interval $0\leq t \leq 6$, what is the average rate of water flow, in gallons per minutes?

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The average value of the rate is given by:

$$\frac{1}{b-a} \int\limits_a^b R(t)   dt$$ $$= \frac{1}{6-0} \int\limits_0^6 3\sqrt{1+3t^3}   dt$$

Evaluating this with a calculator:

$$= \frac{1}{6} \int\limits_0^6 3\sqrt{1+3t^3}   dt \approx \boxed{31.012}$$