It is often said that integral calculus offers a means to solve the area problem. My question, simply aims at understanding what is the interest of this area problem ( at the most basic level).
Suppose that on a graph( with X axis representing time, and Y axis representing speed), I see the straight line y = 50 ( km/h) .
In that case, I understand why I can be interested in knowing what is the area below the curve , say the area enclosed between x = 0 and x = 2 hours). For this area clearly represents the distance that has been travelled in 2 hours.
Are there other elementary/easy cases in which the interest in knowing the measure of the area ( under the curve representing a function) appears as clearly as in this easy case?
Are there other cases in which the measure of the area under the curve reveals itself surprisingly as identical to some quantity we had some prior interest in knowing?