Why is calculus needed in this problem involving work and units of energy? Consider this problem:

Suppose it takes k units of energy to lift a cubic meter of water one meter. About how much energy E will it take to pump dry a circular hole one meter in diameter and 100 meters deep that is filled with water?

Here's how I tried to solve it:
Total volume of water = $\pi r^2h = \pi (\frac{1}{2})^2 100$
Energy required to move this volume of water by $1$ meter $=\pi(\frac{1}{2})^2100k$, and so energy required to move this move this volume of water by $100$ meters $= 100\pi(\frac{1}{2})^2100k = \frac{\pi k10^4}{4}$.
However, the answer given is: $\frac{\pi k10^4}{8}$. Where am I going wrong with my reasoning? Also, this problem was assigned in single variable calculus, under definite integrals. Why is calculus needed here at all?

You don't have to lift every cup of water out by the whole $100$m.  If a given horizontal cross-section of water is a depth $d$ meters, you need to move that cross-section only $d$ meters, not $100$m.  So to find the whole amount of work done you need to add up the work done for each thin cross-section, which depends on the depth; and you can do that with an appropriate integral.
