Work Done by a Spring

Stretching a spring gets harder the farther you pull. Work adds up force times distance in thin slices, which is an integral: the area under the force graph.

How it works

Definite integrals as accumulation

In one sentence

Work is force integrated over distance; for a spring $F = kx$ , so $W = \int_a^b kx\,dx = \tfrac12k\left(b^2 - a^2\right)$ .

General formula

{\htmlData{sym=0}{W}} = \int_{\htmlData{sym=4}{a}}^{\htmlData{sym=5}{b}} {\htmlData{sym=1}{kx}}\,{\htmlData{sym=2}{dx}} = \tfrac12 {\htmlData{sym=3}{k}}\left({\htmlData{sym=5}{b}}^2 - {\htmlData{sym=4}{a}}^2\right)

The force grows as you stretch, so you add up force times tiny distances: the area under the force line.

What the symbols mean

$W$
Work
The energy needed to stretch the spring from a to b.
$kx$
Spring force
Hooke's law: the force grows in proportion to the stretch.
$dx$
Tiny stretch
One thin slice of the stretch. The integral adds up force × distance over all of them.
$k$
Spring constant
How stiff the spring is, in newtons per meter.
$a$
Starting stretch
How far the spring is stretched at the start.
$b$
Final stretch
How far the spring is stretched at the end.

Hooke's law: force grows linearly, $F(x) = kx$ .
Work over a tiny stretch $dx$ is $F(x)\,dx$ . Adding up all the slices is the integral.
The shaded area between $a$ and $b$ is exactly that work.

Learn the math

WorkCalc I · Lesson 53, Applications of Integrals