Heat Shield for a Nose Cone

A rocket's nose cone takes the worst of the heat on the way up, so it's covered in heat-shield material priced by the square meter. Its curved surface is a profile spun around the rocket's axis, and the surface-area integral adds up the thin bands that profile sweeps out.

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

How it works

Surface area of revolution

In one sentence

If $R \ge 0$ has a continuous derivative on $[a, b]$ , revolving its graph about the $x$ -axis sweeps a surface of area $S = 2\pi\int_a^b R(x)\sqrt{1 + R'(x)^2}\,dx$ .

General formula

{\htmlData{sym=0}{S}} = 2\pi\int_0^{\htmlData{sym=3}{{L}}} {\htmlData{sym=1}{R(x)}}\sqrt{1 + {\htmlData{sym=2}{R'(x)}}^2}\,d{\htmlData{sym=4}{x}}

Each tiny piece of the curve, of length $ds$ , sweeps a thin band like a slice of a lampshade: circumference times width, $2\pi R\,ds$ .

What the symbols mean

$S$
Surface area
Square meters of shield needed.
$R(x)$
Profile radius
How far the surface is from the axis at distance x from the tip.
$R'(x)$
Profile slope
How steeply the radius changes along the axis.
$L$
Cone length
3 m from tip to base.
$x$
Position
Distance from the tip along the axis.

The profile is a tangent ogive: an arc of a circle that meets the rocket's body smoothly at the base.
Spinning a short piece of the profile, of length $ds = \sqrt{1 + R'(x)^2}\,dx$ , sweeps out a band of circumference $2\pi R(x)$ .
Each band's area is $2\pi R\,ds$ ; adding them gives the integral.
The $\sqrt{1 + R'^2}$ factor matters near the tip, where the profile is steep: using $dx$ alone would underestimate the area.

Learn the math

Surface AreaCalc II · Lesson 12, Applications of Integrals