Interpretation of definite integrals

Question

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?

Answer 1

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 $x=b$.

Answer 2

Definite integrals are also used to compute probabilities from probability density functions. The nomenclature makes it sound precisely like computing a total Quantity from the density of that Quantity, but PDFs are used to characterize random variables in different situations.

Integrals are also used in determining relative weights of particular harmonics in arbitrary functions. One application you may have heard of is a Fourier Series. In general, integrals are used to define inner products in Hilbert spaces in which such harmonics are defined.