Parametric Arc Length Calculator — Theory
1. Parametric curves
A parametric curve gives the coordinates of a moving point as functions of a parameter \(t\):
\[
x=x(t),\qquad y=y(t).
\]
The point on the curve is
\[
\mathbf r(t)=\langle x(t),y(t)\rangle.
\]
As \(t\) changes from \(a\) to \(b\), the point traces the arc whose length we want to measure.
2. Arc length idea
If the curve is sampled at nearby parameter values, a small piece of the curve has approximate length
\[
\Delta s\approx \sqrt{(\Delta x)^2+(\Delta y)^2}.
\]
In the limit, this becomes an integral.
3. Speed function
Differentiate the coordinates with respect to \(t\):
\[
\frac{dx}{dt}=x'(t),\qquad
\frac{dy}{dt}=y'(t).
\]
The speed of the moving point is
\[
v(t)=\sqrt{\left(x'(t)\right)^2+\left(y'(t)\right)^2}.
\]
This is the rate at which distance along the curve is changing.
4. Parametric arc length formula
The arc length from \(t=a\) to \(t=b\) is
\[
L=\int_a^b
\sqrt{\left(\frac{dx}{dt}\right)^2+
\left(\frac{dy}{dt}\right)^2}\,dt.
\]
Equivalently,
\[
L=\int_a^b v(t)\,dt.
\]
5. Example: \(x=t,\ y=t^2\)
Let
\[
x=t,\qquad y=t^2,\qquad 0\le t\le3.
\]
Differentiate:
\[
\frac{dx}{dt}=1,\qquad
\frac{dy}{dt}=2t.
\]
The speed is
\[
v(t)=\sqrt{1+4t^2}.
\]
Therefore,
\[
L=\int_0^3\sqrt{1+4t^2}\,dt.
\]
An antiderivative is
\[
\int\sqrt{1+4t^2}\,dt
=
\frac{t}{2}\sqrt{1+4t^2}
+
\frac{1}{4}\operatorname{arsinh}(2t).
\]
So
\[
L=
\frac{3}{2}\sqrt{37}
+
\frac{1}{4}\operatorname{arsinh}(6).
\]
Since \(\operatorname{arsinh}(6)=\ln(6+\sqrt{37})\), this may also be written as
\[
L=
\frac{3}{2}\sqrt{37}
+
\frac{1}{4}\ln(6+\sqrt{37}).
\]
6. Circle example
For a circle of radius \(R\),
\[
x=R\cos t,\qquad y=R\sin t.
\]
Then
\[
x'(t)=-R\sin t,\qquad y'(t)=R\cos t.
\]
The speed is
\[
v(t)=\sqrt{R^2\sin^2t+R^2\cos^2t}=R.
\]
Thus, from \(t=a\) to \(t=b\),
\[
L=R|b-a|.
\]
7. Line example
Suppose
\[
x=x_0+\alpha t,\qquad y=y_0+\beta t.
\]
Then
\[
x'(t)=\alpha,\qquad y'(t)=\beta.
\]
The speed is constant:
\[
v(t)=\sqrt{\alpha^2+\beta^2}.
\]
Therefore,
\[
L=\sqrt{\alpha^2+\beta^2}\,|b-a|.
\]
8. Numerical integration fallback
Many parametric curves do not have an elementary antiderivative for their speed function. For example,
ellipses often lead to elliptic integrals. In such cases, numerical integration is the practical method.
The calculator supports:
- Adaptive Simpson: refines where the integrand changes rapidly.
- Composite Simpson: applies Simpson’s rule over many equal subintervals.
- Composite trapezoid: approximates the speed graph with trapezoids.
- Polyline approximation: adds straight-line distances between sampled points.
9. Speed graph interpretation
The speed graph plots \(v(t)\) against \(t\). Since
\[
L=\int_a^b v(t)\,dt,
\]
the area under the speed graph equals the arc length. This is why the calculator shades the accumulated
area during Play mode.
11. Common mistakes
- Using only \(\Delta x\): arc length depends on changes in both \(x\) and \(y\).
- Forgetting the square root: the speed is \(\sqrt{(x')^2+(y')^2}\), not \((x')^2+(y')^2\).
- Using \(dy/dx\) instead of speed: slope and arc length are different ideas.
- Ignoring the interval: the same curve may have different lengths on different \(t\)-intervals.
- Assuming every integral is elementary: many arc length integrals require numerical approximation.
- Forgetting repeated tracing: if the curve is traced more than once, arc length counts the repeated travel.
