Chapter 25 — Key Takeaways

A structured recap of the calculus of paths. The single shift behind everything: a curve is a path, not a graph — let both coordinates depend on a parameter $t$, and the slope, length, and area all follow from the chain rule.

1. Parametric equations and orientation (§25.1–25.2)

A parametric curve is two functions and an interval:

$$x = x(t), \qquad y = y(t), \qquad t \in [a, b].$$

As $t$ increases, the point $(x(t), y(t))$ moves in a definite direction — the orientation. A graph $y = f(x)$ has no orientation; a path always does. This frees us to describe loops, self-crossings, and closed curves (a circle, a cycloid) that no single function can capture. The function case is recovered trivially by $x = t$, $y = f(t)$.

Core parametrizations to know cold:

2. Eliminating the parameter (§25.2)

Erase $t$ algebraically to recover a Cartesian equation when one exists (e.g., $x = R\cos t$, $y = R\sin t \Rightarrow x^2 + y^2 = R^2$; $x = t^2$, $y = t \Rightarrow x = y^2$).

Carry the domain. Eliminating $t$ can silently enlarge the curve. From $x = \cos t$, $y = \cos t$ you get $y = x$ — but only the segment $-1 \le x \le 1$, since $\cos t$ never leaves $[-1, 1]$. After eliminating, always ask which part of the Cartesian curve is actually traced, and in which direction.

Curves that loop or knot (the cycloid, a figure-eight) have no clean Cartesian equation — and that is the whole point of keeping the parametric form.

3. Slope: $dy/dx = \dot y/\dot x$ (§25.3)

$$\frac{dy}{dx} = \frac{dy/dt}{dx/dt} = \frac{\dot y}{\dot x}, \qquad \dot x \neq 0.$$

This is just the chain rule $\dfrac{dy}{dt} = \dfrac{dy}{dx}\cdot\dfrac{dx}{dt}$ solved for $dy/dx$. Tangent classification:

• Horizontal tangent: $\dot y = 0$, $\dot x \neq 0$.
• Vertical tangent: $\dot x = 0$, $\dot y \neq 0$.
• Possible cusp: $\dot x = \dot y = 0$ (the point momentarily stops; e.g., the top of a cycloid arch, the four corners of an astroid).

A self-crossing curve can have two tangent lines at one $(x, y)$ point — one for each $t$ that visits it.

4. Second derivative — the classic trap (§25.3)

$$\frac{d^2y}{dx^2} = \frac{d}{dx}\!\left(\frac{dy}{dx}\right) = \frac{\dfrac{d}{dt}\!\left(\dfrac{dy}{dx}\right)}{\dfrac{dx}{dt}}.$$

The number-one error of this chapter: $\dfrac{d^2y}{dx^2} \neq \dfrac{\ddot y}{\ddot x}$. You may not just differentiate $x$ and $y$ twice and divide. Treat $dy/dx$ as a new function of $t$, differentiate it in $t$, and divide by $\dot x$ — not by $\ddot x$. The sign of $d^2y/dx^2$ then gives concavity as usual.

5. Arc length (§25.4)

$$L = \int_a^b \sqrt{\left(\tfrac{dx}{dt}\right)^2 + \left(\tfrac{dy}{dt}\right)^2}\;dt = \int_a^b \sqrt{\dot x^2 + \dot y^2}\;dt.$$

From the Pythagorean arc-length element $ds = \sqrt{\dot x^2 + \dot y^2}\,dt$. This is the general formula; the Cartesian $L = \int\sqrt{1 + f'(x)^2}\,dx$ of Chapter 18 is the special case $x = t$. Two cautions:

• The curve must be traced exactly once on $[a, b]$, or the integral measures total distance traveled, not geometric length.
• The ellipse perimeter $\int_0^{2\pi}\sqrt{a^2\sin^2 t + b^2\cos^2 t}\,dt$ has no elementary closed form — it is an elliptic integral, evaluated numerically (or via Ramanujan's approximation).

6. Surface area of revolution (§25.5)

Revolving $x(t)$, $y(t)$, $t \in [a, b]$:

$$S_{x\text{-axis}} = \int_a^b 2\pi\,y\,\sqrt{\dot x^2 + \dot y^2}\;dt, \qquad S_{y\text{-axis}} = \int_a^b 2\pi\,x\,\sqrt{\dot x^2 + \dot y^2}\;dt.$$

The factor $2\pi(\text{radius})$ is the circle each point sweeps; $\sqrt{\dot x^2 + \dot y^2}\,dt$ is the band width. Revolving a semicircle recovers the sphere's $4\pi R^2$. This is the integral CAD kernels use to price the surface of any lathe-turned part.

7. Projectile motion (§25.6)

$$x(t) = (v_0\cos\theta)t, \qquad y(t) = (v_0\sin\theta)t - \tfrac12 g t^2.$$

• Path: eliminating $t$ gives a downward parabola (a conic, Chapter 27).
• Time of flight: $T = \dfrac{2v_0\sin\theta}{g}$.
• Range: $R = \dfrac{v_0^2\sin(2\theta)}{g}$; complementary angles $\theta$ and $90^\circ-\theta$ share a range.
• Maximum range at $\theta = 45^\circ$, $R_{\max} = v_0^2/g$ (an optimization result, Chapter 10).

8. The cycloid (§25.7)

The path of a rim point on a wheel of radius $r$ rolling without slipping:

$$x = r(t - \sin t), \qquad y = r(1 - \cos t).$$

• Arc length of one arch: exactly $8r$ — a clean integer multiple, no $\pi$.
• Area under one arch: $3\pi r^2$ — three times the rolling wheel's area.
• Brachistochrone: inverted, it is the curve of fastest descent between two points (beats the straight line).
• Tautochrone: a bead released from rest anywhere on it reaches the bottom in the same time.

These two properties launched the calculus of variations.

9. Three ways to describe a curve (§25.10)

A fluent reader translates freely among them.

10. Connections forward

• Chapter 26 (Polar): the parameter becomes an angle; $x = r(\theta)\cos\theta$, $y = r(\theta)\sin\theta$ is a highly structured parametrization.
• Chapter 27 (Conics): the parabola, ellipse, and hyperbola — including parametric forms like $x = a\cosh t$, $y = b\sinh t$.
• Chapter 28 (Vector-valued functions): everything here rewritten as $\mathbf{r}(t) = \langle x(t), y(t)\rangle$ and lifted to 3D, where velocity, speed, and curvature live.

Reflection

For twenty-four chapters a curve meant a graph — a static rule pairing each $x$ with a height. This chapter overturned that quietly and completely: a curve is a journey, a point moving through the plane and leaving a trail. The mathematics of change grew up from "how one quantity varies with another" into "the study of motion itself." Turn the page, and the parameter becomes an angle.