I need help to understand this problem:
A clepsydra, or water clock, is a glass container with a small hole in the bottom through which water can flow. The "clock" is calibrated for measuring time by placing markings on the container corresponding to water levels at equally spaced times. Let $x=f(y)$ be continuous on the interval $[0,b]$ and assume that the container is formed by rotating the graph of $f$ about the $y$-axis. Let $V$ denote the volume of water and $h$ the height of the water level at time $t$.
a) Determine $V$ as a function of $h$.
By using the disk-method:
$$V=\int_{0}^{h} \pi [f(y)]^2 \,dy=\pi\int_{0}^{h}[f(y)]^2 \,dy$$
b) Show that $\frac{dV}{dt}=\pi[f(h)]^2\frac{dh}{dt}$
I don't understand the proof of this. In the manual solution I have I get this:
$$\frac{dV}{dt}=\frac{dV}{dh}\cdot\frac{dh}{dt}=\pi[f(h)]^2\frac{dh}{dt}$$
I know it is using the Chain Rule. But like that's it? That's all with the proof? And where it comes from the $\pi[f(h)]^2$. It was given in the formula I needed to proof but I want to understand why the $f(h)$.
Thanks in advance for the help.
This is known as the fundamental theorem of calculus.
$$\frac{d}{dh} \int_0^h g(y) \, dy = g(h)$$
Here the $g(y)=f(y)^2$