Foundations — Volume of revolution — disk method, washer method

This page is the toolbox for Volume of Revolution. We assume you have seen nothing. Every symbol the parent note throws at you is unpacked here, in the order that lets each one lean on the one before it.

0. The picture we keep returning to

Look at the figure. A curve $y=f(x)$ sits above a horizontal line (the axis). We take the shaded region between them, grab the axis like a skewer, and spin. The flat region blurs into a solid — here, a bowl/horn shape. Everything below is about naming the parts of this picture precisely.

1. $x$ and $y$ — the two directions

Plain words: $x$ measures how far right you are; $y$ measures how far up.

Picture: two rulers glued at a corner (the origin), one lying flat, one standing up. Any point is "go right $x$, then up $y$".

Why the topic needs it: to spin a region we must first locate it. The axis of revolution is almost always one of these rulers (or a line parallel to them).

2. $f(x)$ — a function, i.e. a "height machine"

Plain words: $f(x)$ is a rule that eats a number $x$ and spits out a height $y$. The letter $f$ is just the machine's name; $x$ is what you feed it.

Picture: stand at position $x$ on the flat ruler, look straight up, and $f(x)$ is how high the curve is directly above you.

Why the topic needs it: the radius of every slice will turn out to be exactly this height. No height machine → no radius → no volume.

$g(x)$ is just a second height machine (a different letter so we can talk about two curves at once — needed for washers).

3. The right triangle nobody mentions — but the radius lives on it

Plain words: the vertical segment from the axis up to the curve is the radius of a slice. Its length is a straight-line distance.

Picture: at position $x$, drop a vertical stick from the point $(x,f(x))$ down to the axis. The stick's length is $f(x)$ when the axis is the x-axis (because "height above $y=0$" is just the $y$-value).

Why "distance", not just "$f(x)$"? When we later spin about a shifted line $y=k$, the stick no longer runs from the curve to zero — it runs from the curve to $k$. Its length becomes $|f(x)-k|$. So the honest definition of radius is distance from axis to curve, and $f(x)$ is only the special case $k=0$.

4. $\pi$ and the area of a circle

Plain words: $\pi \approx 3.14159$ is the fixed number linking a circle's radius to its area and circumference. It never changes.

Picture: take any circle, chop it into thin pizza slices, rearrange them into an almost-rectangle of height $r$ and width $\pi r$ — the rectangle's area $\pi r^2$ is the circle's area.

Why the topic needs it: every slice's flat face is a circle (that's what spinning produces). Its area is the front half of the slice-volume formula.

5. The cylinder — why a thin slice has a known volume

Plain words: a cylinder is a circle dragged sideways a short distance. Its volume is the circle's area times how far you dragged it.

Picture: a coin. Its face is a circle of area $\pi r^2$; its thickness is $h$. Stack the "amount of stuff" = face area $\times$ thickness.

Why the topic needs it: a real slice of the solid has slightly curved sides, so its volume is hard. But if the slice is ultra-thin, the curve barely changes across it, so the slice is almost a perfect cylinder — whose volume we do know. This approximation is the whole trick.

6. $\Delta x$, then $dx$ — "a tiny bit of width"

Plain words: $\Delta x$ (read "delta x") means "a small change in $x$" — the thickness of one slice. $dx$ means the same thing, but shrunk toward zero — an infinitely thin slice.

Picture: the slab's width in the figure above. Make it thinner and thinner; $\Delta x\to dx$.

Why the topic needs it: thickness $h$ in the cylinder formula is this width. Writing $\Delta V\approx \pi [f(x){]}^2\Delta x$ says "this slice's volume ≈ face area × its width". As slices get infinitely thin the approximation becomes exact.

7. $\sum$ and $\int$ — adding up all the slices

Plain words: $\sum$ (capital sigma) means "add these up". $\int$ (the long S) means "add up infinitely many infinitely-thin pieces" — a smooth, exact total.

Picture: finitely many slabs (a chunky staircase estimate) morphing, as they get thinner, into a smooth filled solid. The staircase error vanishes.

Why the topic needs it: one slice gives $\pi r^{2}dx$. The whole solid is the sum of all slices from $x=a$ to $x=b$. Finite sum with $\Delta x$ = a rough guess; the limit as $\Delta x\to 0$ turns $\sum$ into $\int$ and the guess into the truth. This is exactly the Definite Integral as a Limit of Sums machinery.

$a$ and $b$ are the limits of integration — where the region starts and stops along the axis. They are the smallest and largest $x$ of the region.

8. Notation cheat-sheet (what every mark means)

9. Two curves → the washer needs $R$ and $r$

Plain words: if the region floats above the axis with a gap underneath, each spun slice is a ring, not a solid coin — the gap becomes a hole.

Picture: region trapped between an upper curve $f(x)$ and a lower curve $g(x)$. Spin it: the upper curve traces the outer edge (radius $R=f(x)$), the lower curve traces the inner edge of the hole (radius $r=g(x)$).

Why the topic needs it: this is the only new idea in the washer method — which curve is far ($R$) and which is near ($r$). The area is $\pi R^2-\pi r^2$, feeding straight into the integral. Compare with Area Between Curves, where you'd instead subtract heights $f-g$ (no squaring) — different question, different integrand.

Prerequisite map

Equipment checklist

Test yourself — cover the right side.

Connections

• Definite Integral as a Limit of Sums — the $\sum \to \int$ bridge lives here.
• Area Between Curves — same two-curve setup, but you subtract heights, not squared radii.
• Volume by Cross-Sections — disks/washers are the circular special case.
• Shell Method — the alternative that slices parallel to the axis.
• u-substitution — the algebra tool for evaluating shifted-axis integrals.