What situations/models require calculating the area under a curve?

Besides inferring distance traveled from a velocity chart, can anyone name some graphs where you need to know the area under the curve?

For example, I know in Statistics (bell curve), the "probability density function" is used to determine what percentage of the graph is b/w 1, 2, and 3 standard deviations, for example.

Any other big topics (or specific examples) where the area under a graph comes into play?

• Are you asking specifically about finding the area below a curve, or about integrating in general? Because there is an endless list of uses for the latter. – Javier Oct 28 '13 at 19:20
• Ok, sure, integrating in general...Can you list some examples? – JackOfAll Oct 30 '13 at 0:01

You mention distance and velocity. Yes, in this case finding the area under a curve is used. However, this situation can be generalized. The relationship between velocity of distance is that velocity is the first derivative of distance (displacement) with respect to time. Therefore, integrating (finding the area under the curve) of velocity with respect to time gives you change in displacement.

In general, when you have an quantity changing with respect to time - a rate - (or with respect to anything, technically), you would integrate that (area under curve) to find how much the quantity changed.

For example, I could say that that the amount of water in a tank increases at a certain rate (a number of gallons per minute). I could then integrate (find the area under) that curve to find how much the water in the tank increased over the given time.

I could take it further and say that the rate by which the amount of water in the tank is changing, is changing. This is the second derivative (akin to acceleration when talking about displacement) of the amount of water in the tank. Find the area under this curve would tell you how much the rate of water into the tank changed over the given time.

And so on. This is calculus.

• Good reply. Generalizing the position/velocity/acc to a general f(x) = "amount" and f'(x) = "rate of change" and being able to determine one when only knowing the other...Good to know – JackOfAll Oct 30 '13 at 0:00

The "area under a curve" is just the simplest model for a thread of thought fundamental in mathematics as well as in physical, economical, biological, $\ldots\$, environments.

You are given a domain $B\subset{\mathbb R}^n$ (an interval, a disk, a ball, a box, a cylinder, etc.) and some sort of variable "intensity" $$f:\ B\to{\mathbb R}_{\geq0}, \quad x\mapsto f(x)\ .$$ We are interested in the "total effect" caused in this way. This "total effect" will depend on $f$ and on $B$.

If $f$ were constant ($=\eta\>$) on $B$ the "total effect" would be $\eta\cdot{\rm vol}(B)$.

But, alas, the "intensity" $f$ varies from point to point. Nevertheless it makes sense to consider this "total effect of $f$ on $B$" as a reasonable notion and to introduce the notation $$\int\nolimits_B f(x)\ {\rm d}(x)$$ for it. A priori this notion should have the properties \eqalign{\int\nolimits_B \bigl(f(x)+g(x)\bigr)\ {\rm d}(x)&=\int\nolimits_B f(x)\ {\rm d}(x)+\int\nolimits_B g(x)\ {\rm d}(x),\cr \int\nolimits_B \lambda f(x)\ {\rm d}(x)&=\lambda \int\nolimits_B f(x)\ {\rm d}(x)\ ,\cr} and $$\int\nolimits_{A\cup B} f(x)\ {\rm d}(x)=\int\nolimits_A f(x)\ {\rm d}(x)+\int\nolimits_B f(x)\ {\rm d}(x)\ ,$$ when the domains $A$ and $B$ are "essentially disjoint"; and finally $$\int\nolimits_B 1\ {\rm d}(x)={\rm vol}(B)\ .$$

It takes a lot of work to show that when $f$ is continuous on $B$ then for ever finer partitions of $B$ into subdomains $B_k$ $\>(1\leq k\leq N)$ one has $$\int\nolimits_B f(x)\ {\rm d}(x)=\lim_{\ldots}\ \sum_{k=1}^N f(\xi_k)\>{\rm vol}(B_k)\ ,$$ where the $\xi_k\in B_k$ are arbitrary sampling points. On the right hand side of the last formula you can recognize the Riemann sums used to calculate the "area under a curve".