Theory

Density and total quantity

When the function of several variables is a density $\rho (x,y)$ or $\rho (x,y,z)$, then the integral gives the total quantity:

$$\mathrm{Total}={\iint }_{\mathcal{R}}\rho (x,y)dA,\mathrm{Total}={\iiint }_{\mathcal{V}}\rho (x,y,z)dV.$$

If $\rho$ measures the ordinary density, then the integral gives total mass. If $\rho$ measures a charge density, then the integral gives total charge. If $\rho$ measures the probability density, then the integral gives total probability.

Average value

The average value of a distributed quantity is the total amount of that quantity divided by the total measure of the region over which it is spread.

When $f(x,y)$ is defined on a region $\mathcal{R}\subset {\mathbb{R}}^2$, then the average value of $f$ over that region is given by:

$$\bar{f}=\frac{1}{A}{\iint }_{\mathcal{R}}fdA,$$

where $A={\iint }_{\mathcal{R}}dA$. Similarly, if $g(x,y,z)$ is defined on a region $\mathcal{V}\subset {\mathbb{R}}^3$, then the average value of $g$ over that region is given by:

$$\bar{g}=\frac{1}{V}{\iiint }_{\mathcal{V}}gdV,$$

where $V={\iiint }_{\mathcal{V}}dV$.

Center of mass

The center of mass of a planar region or a solid body is the average position of an element of material in the region or body. If the density of the body varies, it is the average position weighted by the density.

Position in a planar region is given by the vector $(x,y)$. Let $A$ be the area of a planar region $\mathcal{R}$. Since integration is linear and respects vector components, we can compute the average position componentwise:

$$\mathbf{CoM}=\frac{1}{A}{\iint }_{\mathcal{R}}(x,y)dxdy=\left(\frac{1}{A}{\iint }_{\mathcal{R}}xdxdy,\frac{1}{A}{\iint }_{\mathcal{R}}ydxdy\right)$$

(and analogously for the $\mathbf{CoM}$ of a solid body, using three components). If there is a varying density, then we need the average weighted by density:

$$\mathbf{CoM}=\frac{1}{M}{\iint }_{\mathcal{R}}\rho (x,y)(x,y)dxdy=\left(\frac{1}{M}{\iint }_{\mathcal{R}}x\rho (x,y)dxdy,\frac{1}{M}{\iint }_{\mathcal{R}}y\rho (x,y)dxdy\right),$$

where $M={\iint }_{\mathcal{R}}\rho (x,y)dxdy$ is now the mass, which will not be proportional to area (or volume) if the density is not uniform.

Moments

Given a quantity $f(x,y,z)$ that depends on points in space, the higher moments of this quantity are given by the integral of the powers of the quantity over the region. These moments are often computed for the case where $f$ returns one of the coordinates, or a radius. For example, when $f=x$ then the first moment is the $x$-coordinate of the center of mass.

An important moment in physics is the moment of inertia, which is the second moment of the distance to the axis of rotation:

$$I={\iiint }_{\mathcal{V}}{d}_{\ell }(x,y,z{)}^2\cdot \rho (x,y,z)dxdydz$$

where $d_{\ell }(x,y,z)$ gives the distance from $(x,y,z)$ to the line $\ell$ that forms the axis of rotation.