1. Using the trapezoidal rule, you get
gives the rate at which the water flows, so if you integrate with respect to time, you get the actual amount of water. In the summation above, the differences between successive 
's form the heights of the trapezoids, while the successive values of form the "bases" of the trapezoids for which you take the averages.
2. Possibly... We know that is differentiable, so 
certainly exists and must be continuous. The intermediate value theorem says that, on the interval ![[0,8]](https://tex.z-dn.net/?f=%5B0%2C8%5D)
, we can find some 
such that 
would fall between 
and 
. But from the given data, we can't guarantee that is ever 0, because both 
and 
.
3. The average rate of water flow would be
4.
is increasing whenever 
. By the fundamental theorem of calculus,
and
for 
only.
5. For
, we have 
, so
2. Possibly... We know that
3. The average rate of water flow would be
4.
and
5. For