I came across this problem in School. Suppose I have a cup and suppose that the cup is cylinder. I know the total volume of the cup and I fill with some water.
Let $V_c$ be the volume of the cup, $V_w$ the volume of water and $h$ be the distance from the water to the bottom of the cup. I'd like to calculate the total volume of water inside the cup (the cup is not filled).
Here's what I thought: since the volume is proportional to the height I could do a basic cross multiplication, am I correct? Is there a way to write that result using integral?
You need to know the height of the cup $-$ call it $H$. Then the answer is simply $V_w=\dfrac{h}{H}V_c$.
You don't need to use integration here, because the shape of the cup is so simple. If it had a more complicated shape, you might need calculus.