Let $r(x, y, t)$ be the rate of rainfall (in cm/day) at the location $(x, y)$ (measured in km), $t$ days after Jan 1st, 2017. Let $R$ be the domain defined as California × [0, 365]. What are the units of $\int\int\int_ R r(x, y, t) dxdydt$, and what does that quantity represent? To find the average value of $r(x, y, t)$ over that domain, we would need to divide this integral by a constant, the “volume” of $R$. What are the units of this “volume”? What are the units of the average value?
Is my understand correct in that the units for the first question is $(cm/day)(km)(km)(day)=cm*km^2$ and the units for the second question is $(km)(km)(day) = km^2 * day$? (For first question I did units of $r(x,y,t)$ times units of $x$ times units of $y$ times units of $t$, and for the second I did units of $x$ times units of $y$ times units of $t$). I am not quite sure, however, what the units would be for the average value.
Note that this is length^3, or volume. So you can simplify the units by picking one unit to work with; for instance, you could divide your answer by 100,000 and the result would be cubic km. Or you could multiply it by 10,000,000 and the result would be liters.
Yes.
The units for an average value are the same as the original. You have the units of cm∗km$^2$ divided by units of km$^2$∗day, so that cancels out to cm/day.