Arc Length of an Archimedean Spiral

Why the Archimedean Spiral Has a Manageable Arc-Length Integral

The Archimedean spiral \(r=a+b\theta\) changes its radius at a constant rate as the angle changes. Its shape can wind around the pole, but its arc length over a finite angular interval can be found by reducing the polar arc-length integral to an ordinary integral of the form \(\int\sqrt{u^2+c^2}\,du\).

The key is to keep track of the rate \(b\). It affects both the speed along the curve and the change of variable used to evaluate the integral. In particular, replacing \(dr/d\theta\) by \(1\) is only valid when \(b=1\).

From the Polar Formula to the Spiral Integral

For a polar curve \(r=r(\theta)\) on an interval \([\theta_0,\theta_1]\), where \(\theta_0\leq\theta_1\), the arc length is

This formula follows by writing the curve in Cartesian coordinates as \(x=r\cos\theta\) and \(y=r\sin\theta\). Differentiating gives \(x'=r'\cos\theta-r\sin\theta\) and \(y'=r'\sin\theta+r\cos\theta\). Squaring and adding cancels the cross terms:

The speed is therefore \(\sqrt{(x')^2+(y')^2}=\sqrt{r^2+(r')^2}\), which gives the polar arc-length formula.

For the Archimedean spiral \(r=a+b\theta\), the derivative is \(dr/d\theta=b\). Thus, when \(b\neq0\), its length over a finite interval is

The integral is finite on every finite interval because its integrand is continuous there. If \(b=0\), then \(r=a\) is constant and the speed is \(|a|\), so the length is \(|a|(\theta_1-\theta_0)\). The rest of the evaluation focuses on the nonconstant case \(b\neq0\).

Substitution and the Sign of \(b\)

Set \(u=a+b\theta\). Then \(du=b\,d\theta\), and the endpoint values become \(u_0=a+b\theta_0\) and \(u_1=a+b\theta_1\). The integral becomes

This expression also works when \(b<0\). In that case \(u_1<u_0\), so the integral with respect to \(u\) is negative, as is \(1/b\); their product is positive. Equivalently, the positive length can be written using an antiderivative \(F\) as \(\bigl(F(u_1)-F(u_0)\bigr)/b\).

The Standard Antiderivative

For a fixed nonzero real number \(c\), an antiderivative of \(\sqrt{u^2+c^2}\) is

The logarithm is defined for every real \(u\) because \(\sqrt{u^2+c^2}>|u|\) when \(c\neq0\), and hence \(u+\sqrt{u^2+c^2}>0\). Here is a derivation, so the logarithmic term and its coefficient are accounted for rather than memorized without justification.

For the spiral, take \(c=b\); the antiderivative depends on \(b^2\), so its sign does not change when \(b\) is negative. The arc-length formula is therefore

As a check on the derivation, differentiating \(F_b\) gives \(\sqrt{u^2+b^2}\). Consequently, differentiating \(F_b(a+b\theta)/b\) with respect to \(\theta\) gives \(\sqrt{(a+b\theta)^2+b^2}\), the original speed.

Worked Examples

A Common Error: Losing the Radial Rate

A frequent mistake is to use \(1\) in the speed formula in place of \((dr/d\theta)^2\), even when the radius does not have derivative \(1\). For example, for \(r=1+2\theta\), an incorrect setup is

This calculation uses the wrong derivative: \(dr/d\theta=2\), not \(1\). The correct setup is

The correction affects both parts of the substitution: the constant inside the square root is \(b^2\), and \(d\theta=du/b\). Keeping these linked prevents a wrong speed or a missing factor in the final integral.

Why the Method Is Useful

The substitution works because the spiral's radius is a linear function of the angle. It turns the changing radius \(a+b\theta\) into the integration variable \(u\), while the constant radial rate contributes the fixed term \(b^2\) under the square root. The geometry is thus reduced to one standard integral, regardless of the starting radius or whether the spiral moves outward or inward as \(\theta\) increases.

The formula also offers a reliable way to check a calculation: differentiate the proposed antiderivative with the chain rule and compare the result with the polar speed. When \(b<0\), check the transformed endpoints as well as the sign of \(1/b\). These checks address the two common sources of errors: omitting the radial rate and reversing the substitution inconsistently.