examples of cases showing that knowing the area under a curve really matters ( at the elementary level)

Question

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).

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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?

Answer 1

Needing to knowing the area under a curve pops up all over the place in physics, though most examples can hardly be considered elementary in the sense that you are looking for. Perhaps the simplest example is that of work. Under a constant force $F$, the amount of work $W$ done on an object over distance $\Delta s$ is given by $$ W=F\Delta s $$ However, if we instead let the force vary (as it often does in the real world), we instead need to examine the area under the curve described by the force vs. distance, i.e. $$ W=\int F(s)\, \mathrm ds $$

Answer 2

Because, for intance, the area of the circle centered at $(0,0)$ with radius $r>0$ is twice the area under the curve $\sqrt{r^2-x^2}$ ($x\in[-r,r]$).

Answer 3

Area under the curve arises in e.g. calculating probabilities.

If you have a probability density function $f(x)$ for a continuous random variable $X$ then the following is true $$ P(a \lt X < b)= \int_{a}^{b}f(x)\,dx $$ $X$ may represent the time one waits to receive one answer to a question posted on StackExchange, or the actual weight of a burger advertised as a 'quarter pounder' or the actual volume of liquid in a 'one pint' drink, or household income, or scores in an exam etc. etc.