Applications to Physics, Economics, and Probability

Hydrostatic force, centers of mass, consumer surplus, and probability all reduce to one construction: cut a quantity into small pieces, approximate each, sum, integrate. Only the setting changes. The pieces are strips of a submerged plate, point masses on a lamina, dollars saved at a price, and slices of probability.

Hydrostatic pressure and force

Water pressure grows with depth, because the weight of the fluid above a point increases. A thin horizontal plate of area $A$ at depth $d$ in a fluid of density $\rho$ supports a column of fluid of weight $\rho gAd$, so the pressure (force per unit area) is

A key experimental fact makes the definite integral usable: at a fixed depth, fluid pressure acts equally in all directions. So the pressure on a vertical surface at depth $d$ is also $\rho gd$. This lets the force on a vertical wall or dam be computed even though the pressure is not constant over the wall.

The wall is cut into thin horizontal strips. A strip at depth $x$ has nearly constant pressure $\rho gx$ and area $w(x)\Delta x$, where $w(x)$ is the strip's width, so its force is about $\rho gxw(x)\Delta x$.

Moments and centers of mass

A plate balances horizontally on a single point: its center of mass. The one-dimensional law is Archimedes' Law of the Lever: two masses on a rod balance about the fulcrum when $m_1{d}_1=m_2{d}_2$.

Placing masses $m_i$ at coordinates $x_i$ on the $x$-axis, the balance point is the mass-weighted average

where $M$ is the moment about the origin and $m$ the total mass. In the plane, masses $m_i$ at $(x_i,y_i)$ have two moments, one about each axis:

• Moment about the $y$-axis: $M_y={\sum }_i{m}_i{x}_i$ (measures the tendency to rotate about the $y$-axis).
• Moment about the $x$-axis: $M_x={\sum }_i{m}_i{y}_i$.

The center of mass is $(\bar{x},\bar{y})=(M_y/m,M_x/m)$ — the point where a single particle of mass $m$ would have the same moments as the whole system.

The centroid of a lamina

For a flat plate of uniform density occupying a region under $y=f(x)$ over $[a,b]$, replace the discrete masses by strips. A strip at $x$ has mass proportional to its area $f(x)\Delta x$, and its own center of mass sits at its midpoint height $\frac{1}{2}f(x)$. Summing the strip moments and passing to the limit, the density cancels, and the balance point — the centroid — depends only on the shape.

The Theorem of Pappus

Once the centroid is known, a volume of revolution follows without an integral.

Economics: consumer and producer surplus

The demand function $p(x)$ gives the price at which $x$ units sell; it decreases, since selling more requires a lower price. At the market price $P=p(X)$, consumers who would have paid more than $P$ keep the difference. Cut $[0,X]$ into intervals: the buyers of the units near $x_i$ valued them at $p(x_i)$ but paid only $P$, saving $[p(x_i)-P]\Delta x$. Summing gives the total saving.

The mirror image on the supply side is producer surplus, ${\int }_0^X[P-p_S(x)]\mathbf{d}x$, the extra earned by producers willing to sell below $P$. Their sum, the total surplus, is maximized when supply meets demand at the equilibrium price.

The same slicing measures biological flow. Blood in a vessel of radius $R$ moves fastest at the center: $v(r)=\frac{P}{4\eta l}(R^2-r^2)$ at distance $r$ from the axis.

Probability

A continuous random variable $X$ — a height, a lifetime, a waiting time — takes values across an interval, and probabilities attach to intervals, not points. They are areas under a probability density function.

The first condition says total probability is $1$; the second reads the probability of any interval as the area over it.

The mean

Balancing the density like a lamina locates its center. The mean is the $x$-coordinate of the centroid of the region under $f$ — the value about which $f$ balances.

Waiting times and failure times are commonly modeled by the exponential density $f(t)=c{e}^{-ct}$ for $t\ge 0$ (and $0$ for $t<0$).

The normal distribution. Heights, test scores, and measurement errors follow the bell-shaped normal density

with mean $\mu$ and standard deviation $\sigma$ controlling the spread. The integrand has no elementary antiderivative, so interval probabilities are found numerically — the reason the normal table exists.

The common construction

The same integral recurs across every quantity above, differing only in the contribution named for a single piece.