Volume of a Rocket Nozzle

A rocket nozzle is a curve spun around an axis. Slice it into thin washers, rings with a hole, and add their volumes: that's the washer method. As the washers get thinner, the sum becomes an integral and gives the exact amount of metal.

Drag to turn the solid. The amber region is what gets revolved.

How it works

Volumes by disks and washers

In one sentence

If the region between $y = r(x)$ and $y = R(x)$ ( $0 \le r \le R$ ) on $[a, b]$ is revolved about the $x$ -axis, the solid's volume is $V = \pi\int_a^b \left(R(x)^2 - r(x)^2\right)dx$ .

General formula

{\htmlData{sym=0}{V}} = \pi\int_a^b \left({\htmlData{sym=1}{R(x)}}^2 - {\htmlData{sym=2}{r(x)}}^2\right){\htmlData{sym=3}{dx}}

Each slice is a coin with a hole. Area of the coin minus area of the hole, times thickness, is its volume; the integral adds all the coins.

What the symbols mean

$V$
Volume
The amount of metal in the nozzle wall.
$R(x)$
Outer radius
Distance from the axis to the outside surface at x.
$r(x)$
Inner radius
Distance to the inside surface: the hole.
$dx$
Slice thickness
How thick each washer is.

Spin the region between the outer wall $R(x)$ and the inner wall $r(x)$ around the axis.
A thin slice at $x$ is a washer: a disk of radius $R(x)$ minus a hole of radius $r(x)$ , thickness $dx$ .
Its volume is $\pi(R^2 - r^2)\,dx$ . Add them: a Riemann sum.
The limit is the integral above, the exact volume of the nozzle's wall.

Learn the math

Volumes by Disks and WashersCalc I · Lesson 50, Applications of Integrals