Calibrating a Fuel Gauge on a Horizontal Tank

A cylindrical tank lying on its side is narrow at the bottom, wide in the middle and narrow at the top, so the fuel volume doesn't rise evenly with depth. The cross-section is an integral of a square root, √(r² − y²), and trig substitution turns it into something you can integrate.

How it works

Trig substitution

In one sentence

For integrands with $\sqrt{a^2 - x^2}$ , substitute $x = a\sin\theta$ (so $\sqrt{a^2 - x^2} = a\cos\theta$ and $dx = a\cos\theta\,d\theta$ ); for $\sqrt{a^2 + x^2}$ use $x = a\tan\theta$ , and for $\sqrt{x^2 - a^2}$ use $x = a\sec\theta$ .

General formula

{\htmlData{sym=0}{A(h)}} = \int_{-{\htmlData{sym=2}{r}}}^{{\htmlData{sym=1}{h}}-{\htmlData{sym=2}{r}}} 2\sqrt{{\htmlData{sym=2}{r}}^2 - {\htmlData{sym=3}{y}}^2}\,d{\htmlData{sym=3}{y}} \;\overset{{\htmlData{sym=3}{y}} = {\htmlData{sym=2}{r}}\sin{\htmlData{sym=4}{\theta}}}{=}\; {\htmlData{sym=2}{r}}^2\cos^{-1}\!\left(\frac{{\htmlData{sym=2}{r}}-{\htmlData{sym=1}{h}}}{\htmlData{sym=2}{{r}}}\right) - ({\htmlData{sym=2}{r}}-{\htmlData{sym=1}{h}})\sqrt{2{\htmlData{sym=2}{r}}{\htmlData{sym=1}{h}} - {\htmlData{sym=1}{h}}^2}

The substitution walks around the circle by angle instead of by height, and the square root that measured the circle's width becomes a plain cosine.

What the symbols mean

$A(h)$
Fuel cross-section
Area of the tank's round end covered by fuel of depth h.
$h$
Fuel depth
What the dipstick reads.
$r$
Tank radius
Half the tank's diameter.
$y$
Height from center
Where a thin horizontal slice of fuel sits.
$\theta$
Substitution angle
y = r sin θ walks around the circle by angle.

Slice the tank's round end at height $y$ (measured from its center): the fuel there is $2\sqrt{r^2 - y^2}$ wide.
Substitute $y = r\sin\theta$ : the root becomes $r\cos\theta$ and $dy$ becomes $r\cos\theta\,d\theta$ .
The integral is then $r^2\int 2\cos^2\theta\,d\theta = r^2(\theta + \sin\theta\cos\theta)$ , which gives the segment's area.
Multiply by the tank's length for the volume. The gauge's marks follow this curve, not a straight line.

Learn the math

Trig SubstitutionsCalc II · Lesson 3, Integration Techniques Integration StrategyCalc II · Lesson 7, Integration Techniques