7.2 Calculus of Parametric Curves

Parametric Curves

Parametric curves describe paths in two dimensions by defining $x$ and $y$ as separate functions of a parameter $t$. Instead of writing $y$ as a function of $x$, you let both coordinates depend on $t$, which makes it possible to represent motions and shapes that a single $y = f(x)$ equation can't capture.

With this setup, the standard calculus toolkit extends naturally. You can find slopes, enclosed areas, arc lengths, and surface areas of revolution, all through formulas built around $x(t)$ and $y(t)$.

Derivatives of Parametric Curves

The slope of a parametric curve at a point isn't found by differentiating $y$ with respect to $x$ directly. Instead, you use the chain rule. Since both $x$ and $y$ depend on $t$, the derivative $\frac{dy}{dx}$ comes from dividing the two rates:

$\frac{dy}{dx} = \frac{dy/dt}{dx/dt} = \frac{y'(t)}{x'(t)}$

This works whenever $x'(t) \neq 0$.

Once you have the slope at a specific parameter value $t_0$, the tangent line equation at the point $(x(t_0),\, y(t_0))$ is:

$y - y(t_0) = \frac{y'(t_0)}{x'(t_0)}\big(x - x(t_0)\big)$

Finding horizontal and vertical tangent lines:

• Horizontal tangent: Set $y'(t) = 0$ (the numerator is zero) while confirming $x'(t) \neq 0$. For example, a cycloid $x = t - \sin t,\; y = 1 - \cos t$ has horizontal tangents where $y'(t) = \sin t = 0$.
• Vertical tangent: Set $x'(t) = 0$ (the denominator is zero) while confirming $y'(t) \neq 0$.
• If both $x'(t) = 0$ and $y'(t) = 0$ at the same $t$, you have an indeterminate case and need to investigate further (often using L'Hôpital's Rule on the ratio).

For the second derivative, which you'll need for concavity and curvature:

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

Differentiate $\frac{dy}{dx}$ with respect to $t$ first, then divide by $x'(t)$. A common mistake is to just compute $\frac{y''(t)}{x''(t)}$, which is not the same thing.

Area Within Parametric Curves

To find the area under a parametric curve (between the curve and the $x$-axis), substitute the parametric equations into the standard area integral. The formula is:

$A = \int_{t_1}^{t_2} y(t)\, x'(t)\, dt$

This comes from replacing $y$ with $y(t)$ and $dx$ with $x'(t)\, dt$ in the integral $\int y\, dx$.

A few things to watch for:

• Direction of tracing matters. The limits $t_1$ and $t_2$ should correspond to the curve being traced in the direction that keeps the enclosed region on the correct side. If the curve traces left to right as $t$ increases, you get positive area directly. If it traces right to left, you may need to negate the integral or swap limits.
• Simple closed curves. The formula works cleanly when the curve is traced exactly once. If the parametrization traces the curve multiple times over your interval, you'll overcount the area.
• Signed area. The integral can give a negative value depending on orientation. Take the absolute value if you need the geometric area.

Arc Length of Parametric Curves

The arc length of a parametric curve from $t = a$ to $t = b$ is:

$L = \int_a^b \sqrt{\big(x'(t)\big)^2 + \big(y'(t)\big)^2}\, dt$

The intuition here is the Pythagorean theorem applied to infinitesimal pieces of the curve. Over a tiny change $dt$, the curve moves $x'(t)\, dt$ horizontally and $y'(t)\, dt$ vertically, so the tiny arc length is $\sqrt{(x'\, dt)^2 + (y'\, dt)^2} = \sqrt{x'^2 + y'^2}\, dt$.

Steps to compute arc length:

1. Find $x'(t)$ and $y'(t)$.
2. Square each, add them, and simplify the expression under the radical. (Algebraic simplification here can save you a lot of pain in the integration step.)
3. Integrate from $a$ to $b$.

Many arc length integrals don't simplify to elementary functions, so don't be surprised if a problem asks you to set up the integral without evaluating it, or to use numerical methods.

Surface Area from Parametric Revolution

When you revolve a parametric curve around an axis, the surface area formulas mirror the arc length formula but include a factor for the radius of revolution.

Revolution around the $x$-axis (radius of revolution is $y(t)$):

$S = 2\pi \int_a^b y(t)\,\sqrt{\big(x'(t)\big)^2 + \big(y'(t)\big)^2}\, dt$

Revolution around the $y$-axis (radius of revolution is $x(t)$):

$S = 2\pi \int_a^b x(t)\,\sqrt{\big(x'(t)\big)^2 + \big(y'(t)\big)^2}\, dt$

The structure is the same in both cases: $2\pi \times (\text{radius}) \times (\text{arc length element})$. The only difference is whether $y(t)$ or $x(t)$ serves as the distance from the axis.

Steps to set up a surface area integral:

1. Identify which axis the curve revolves around.
2. Write down the appropriate radius function ($y(t)$ for $x$-axis, $x(t)$ for $y$-axis). Make sure this quantity is non-negative over the interval.
3. Compute $x'(t)$ and $y'(t)$, then form the arc length element $\sqrt{x'^2 + y'^2}$.
4. Determine the correct limits of integration from the parametrization.
5. Assemble and evaluate the integral.

Vector-Valued Functions and Curvature

A parametric curve $x(t),\, y(t)$ can be written as a vector-valued function:

$\mathbf{r}(t) = \langle x(t),\, y(t) \rangle$

This notation is compact and connects directly to the ideas of velocity and acceleration. The derivative $\mathbf{r}'(t) = \langle x'(t),\, y'(t) \rangle$ gives the tangent (velocity) vector at each point.

Curvature measures how sharply a curve bends at a given point. For a parametric curve, the formula is:

$\kappa = \frac{|x'(t)\, y''(t) - y'(t)\, x''(t)|}{\big(x'(t)^2 + y'(t)^2\big)^{3/2}}$

• A straight line has $\kappa = 0$ everywhere (no bending).
• A circle of radius $r$ has constant curvature $\kappa = 1/r$. Smaller circles curve more sharply.

The unit normal vector $\mathbf{N}$ is perpendicular to the tangent vector and points toward the center of curvature (the center of the "best-fit circle" at that point). You find it by first computing the unit tangent vector $\mathbf{T} = \frac{\mathbf{r}'(t)}{|\mathbf{r}'(t)|}$, then taking $\mathbf{N} = \frac{\mathbf{T}'(t)}{|\mathbf{T}'(t)|}$.

Applications and Examples

Parametric equations show up whenever position depends on time or another independent variable.

• Projectile motion: A projectile launched at speed $v_0$ and angle $\theta$ has $x(t) = v_0 \cos\theta\, t$ and $y(t) = v_0 \sin\theta\, t - \frac{1}{2}g t^2$. The arc length formula gives the total distance along the trajectory, which is longer than the horizontal range.
• Planetary orbits: An elliptical orbit can be parametrized as $x(t) = a\cos t,\; y(t) = b\sin t$. The area formula lets you compute the area swept out over a portion of the orbit, connecting to Kepler's second law.
• Cycloids: The path traced by a point on the rim of a rolling circle is $x(t) = r(t - \sin t),\; y(t) = r(1 - \cos t)$. This classic curve has interesting properties: its arc length over one full arch works out to exactly $8r$.

General problem-solving approach:

1. Write down (or identify) the parametric equations $x(t)$ and $y(t)$.
2. Determine what the problem asks for: slope, area, arc length, or surface area.
3. Choose the corresponding formula.
4. Compute the necessary derivatives $x'(t)$ and $y'(t)$.
5. Set up the integral with correct limits (check what values of $t$ correspond to the portion of the curve you care about).
6. Evaluate the integral, simplifying the integrand first when possible.

Some curves have natural connections to polar coordinates. For instance, a spiral $r = e^{\theta}$ in polar form can be converted to parametric form via $x(\theta) = e^{\theta}\cos\theta,\; y(\theta) = e^{\theta}\sin\theta$, and then all the parametric calculus tools apply.