Interpretation of definite integrals [closed]

It is known from the Fundamental Theorem of Calculus that $$\int_a^b f(x)=F(b)-F(a).$$ This has the geometric interpretation of the net area between $f(x)$ and the $x-$axis. I suspect that there are many other interpretations for the result.

Some other examples of definite integration which I have found are:

• Volumes in higher dimensions: $\iint_R f(x) dV$
• Applications in kinematics: e.g. $distance=\int_0^{t_0}v(t)\,dt$, where $v(t)$ is the speed of the object
• Calculation of volumes in solids of revolution: $\pi\int_a^b f^2(x)\,dx$
• Calculation of fluxes (surface integrals): $\iint_S f\,dS$
• Calculation of line integrals: $\int_C f\,ds$
• Arc length of a curve: $\int_a^b \sqrt{1+f'(x)^2}\,dx$

What other interpretations are there for a definite integral? And are the interpretations specific to one application?

closed as too broad by Grigory M, egreg, hardmath, Thomas Andrews, apnortonDec 17 '13 at 22:21

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• There are about one gazillion applications of definite integrals. – mrf Jan 17 '13 at 9:05

Whenever $f(t)$ represents a rate of change of something, $\int_a^b f(t)\ dt$ represents the total change from $t=a$ to $t=b$.
If $f(x)$ represents a density of something, $\int_a^b f(x)\ dx$ represents the total amount of that thing from $x=a$ to $x=b$.