Chapter 18 — Key Takeaways
The one idea: slice and sum
Every application in this chapter is the same four-step recipe (§18.1) in a different costume:
- Draw the picture and pick a slicing variable ($x$, $y$, or $t$).
- Cut a thin slice at a general position, of thickness $dx$ (or $dy$, $dt$).
- Write the slice's contribution — $dA$, $dV$, $dW$, $dF$, or $dM$ — as a function of the variable times the thickness.
- Integrate over the full range.
The setup is the hard part; the integration is mechanical. Steps 1–3 are where every error and every insight live. Step 4 draws on the techniques of Chapters 15–16, or on a computer. This chapter is really a course in modeling: turning a picture into a single integrand.
The formula table (§18.11)
| Quantity | Slice contribution | Integral | Section |
|---|---|---|---|
| Area between curves | $[f - g]\,dx$ (top − bottom) | $\displaystyle\int_a^b (f - g)\,dx$ | §18.2 |
| Volume — disks ($\perp$ axis) | $\pi[f]^2\,dx$ | $\displaystyle\int_a^b \pi[f]^2\,dx$ | §18.3 |
| Volume — washers ($\perp$ axis) | $\pi(f^2 - g^2)\,dx$ | $\displaystyle\int_a^b \pi(f^2 - g^2)\,dx$ | §18.3 |
| Volume — shells ($\parallel$ axis) | $2\pi x\,f\,dx$ | $\displaystyle\int_a^b 2\pi x\,f\,dx$ | §18.4 |
| Arc length | $\sqrt{1 + (f')^2}\,dx$ | $\displaystyle\int_a^b \sqrt{1 + (f')^2}\,dx$ | §18.5 |
| Surface of revolution | $2\pi f\sqrt{1 + (f')^2}\,dx$ | $\displaystyle\int_a^b 2\pi f\sqrt{1 + (f')^2}\,dx$ | §18.6 |
| Work (variable force) | $F(x)\,dx$ | $\displaystyle\int_a^b F(x)\,dx$ | §18.7 |
| Hydrostatic force | $\rho g h\,w(h)\,dh$ | $\displaystyle\int_a^b \rho g h\,w(h)\,dh$ | §18.8 |
| Center of mass (rod) | $x\,\rho(x)\,dx$ over $\rho(x)\,dx$ | $\displaystyle\bar x = \frac{\int_a^b x\rho\,dx}{\int_a^b \rho\,dx}$ | §18.9 |
Area between curves (§18.2)
- Always integrate top minus bottom (or right minus left for horizontal strips). The formula ignores whether the curves sit above or below the axis — only the gap matters.
- If the curves cross inside $[a,b]$, split at every crossing. A single signed integral lets negative pieces cancel positive area, giving a wrong (sometimes negative) answer. Area is always non-negative.
- Choosing vertical vs. horizontal strips to avoid splitting is the central setup decision. A region bounded by $x = y^2$ and $x = y + 2$ needs one horizontal integral but two vertical ones.
Disk/washer vs. shell — the decision (§§18.3–18.4)
- Disks/washers slice perpendicular to the axis; the slice is a coin (disk) or coin-with-a-hole (washer). Washer area is $\pi(f^2 - g^2)$, not $\pi(f - g)^2$ — square each radius then subtract.
- Shells slice parallel to the axis; the slice is a tin can of radius "distance to axis," height $f$, thickness $dx$, contributing $2\pi(\text{radius})(\text{height})\,dx$.
- How to choose: slice perpendicular when the region is a function of the same variable as the axis; slice parallel (shells) when it is a function of the other variable. Shells avoid having to invert $y = f(x)$ into $x = g(y)$.
- Both methods give the same volume — it is a convenience choice, not a correctness one.
- A shifted axis changes only the radius: distance from the slice to the new axis (e.g. $2 - x$ to rotate about $x = 2$, or $x + 1$ about $x = -1$).
Arc length and surface area (§§18.5–18.6)
- The arc-length element $ds = \sqrt{1 + (f')^2}\,dx$ is the hypotenuse of a tiny right triangle with legs $dx$ and $dy = f'(x)\,dx$. It reappears in surface area and in line integrals (Ch. 35).
- Surface area uses the slant element $ds$, not the flat $dx$. Forgetting the radical computes a stack of flat rings and undercounts a tilted surface.
- Most arc-length integrals are non-elementary. Even $y = x^2$ needs trig substitution (Ch. 16); the ellipse perimeter is the famous elliptic integral with no elementary form. When the antiderivative does not exist, integrate numerically (§18.10).
Work (§18.7)
- $W = \int_a^b F(x)\,dx$ extends $W = F \cdot d$ to a force that varies with position.
- Springs: $W = \tfrac12 k L^2$ — energy grows with the square of the stretch.
- Pumping: slice the fluid into horizontal slabs; $dW = (\text{slab weight}) \times (\text{lift distance})$. Two distances are in play — the slab's position and its lift distance. If $y$ is measured from the bottom and water exits at height $H$, the lift distance is $H - y$, not $y$.
- Cables/chains: the still-hanging portion shrinks as you lift; integrate its varying weight over the lift.
Hydrostatic force (§18.8)
- $F = \int \rho g\,h\,w(h)\,dh$, slicing the plate into horizontal strips at depth $h$.
- Force is not pressure times total area: pressure $\rho g h$ varies with depth and cannot leave the integral. The width $w(h)$ also varies for non-rectangular plates — read it off the geometry first.
- The center of pressure (force-weighted average depth) is a second moment integral and always lies below the area centroid, because pressure piles up with depth.
Center of mass (§18.9)
- $\bar x = M_O / M = \int x\rho\,dx \,/\, \int \rho\,dx$ — total moment over total mass, the continuous version of a weighted average.
- For a uniform 2D plate the density cancels and you get the centroid, a purely geometric point.
- Pappus's theorem links §§18.3–18.4 to §18.9: revolving a region of area $A$ about an external axis sweeps volume $V = 2\pi\bar d\,A$, where $\bar d$ is the centroid's distance to the axis (a torus gives $2\pi^2 R r^2$ instantly).
- The same moment formula is the definition of expected value for a probability density — $\mu = \int x\,p(x)\,dx$.
Common errors to avoid
- Using a single signed integral for area when curves cross — split at crossings.
- Writing the washer integrand as $\pi(f - g)^2$ instead of $\pi(f^2 - g^2)$.
- Dropping the $\pi$ in volumes, or the radical $\sqrt{1 + (f')^2}$ in surface area.
- In pumping problems, using the slab's position as its lift distance.
- Treating hydrostatic force as (pressure) × (area) — pressure stays inside the integral.
- Forgetting to divide by total mass: $\bar x = M_O/M$, not just $\int x\rho\,dx$.
- Confusing force $F(x)$ with work $\int F(x)\,dx$ — different quantities, different units.
Connections
- Chapters 15–16 supply the integration techniques (substitution, parts, trig substitution) that evaluate these integrands.
- Chapter 17 (improper integrals) handles infinite domains, e.g. the probability density in Exercise 18.40 and the gravitational-work limit.
- Chapter 19 (differential equations) is the next application of integration — from static accumulations here to dynamic systems.
- Chapters 25–26 generalize arc length to parametric and polar curves.
- Chapter 32 lifts the whole slice-and-sum machine into double and triple integrals: volumes of arbitrary 3D solids (not just solids of revolution) and centers of mass of arbitrary bodies. Everything here is the one-variable warm-up for that.
- Chapter 35 reuses the arc-length element $ds$ in line integrals.
A final reflection
The "slice and sum" pattern is one of the great unifying ideas of calculus. Internalize it and you can derive the volume of a cone, the surface area of a sphere, the work to drain a tank, or the force on a flood gate — without memorizing a single one of those formulas. You identify the slice, write its contribution, and integrate. That generative power — a handful of mental moves producing an unlimited variety of correct results — is worth far more than any one formula in the table above. It is the foundation of every quantitative field, and you carry it into Chapter 19 and beyond.