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://img.qammunity.org/2019/formulas/mathematics/college/g5vpsbhqtnv0av2lkaf05414po2a3c6rrl.png)
, 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



