Calculus of Parametric Curves: Comprehensive Study Guide

This guide covers the application of differential and integral calculus to curves defined by parametric equations. Instead of treating $y$ as a direct function of $x$ , we analyze both as functions of an independent parameter, typically $t$ .

Learning Objectives

After studying this material, you should be able to:

1.2.1 Determine derivatives and equations of tangents for parametric curves.
1.2.2 Find the area under a parametric curve.
1.2.3 Use the equation for arc length of a parametric curve.
1.2.4 Apply the formula for surface area to a volume generated by a parametric curve.

Key Terms & Glossary

Parameter ( $t$ ): An independent variable that determines the coordinates $(x, y)$ of a curve simultaneously.
Parametric Curve: A set of points $(x(t), y(t))$ generated as the parameter $t$ varies over an interval.
Tangent Vector: A vector representing the instantaneous direction of motion along a parametric curve.
Smooth Curve: A curve where $x'(t)$ and $y'(t)$ are continuous and not simultaneously zero.

The "Big Idea"

In standard Cartesian calculus, we describe what a path looks like ( $y=f(x)$ ). In parametric calculus, we describe how a path is traversed. This allows us to model motion (like a baseball's trajectory) where time is the driving factor, and to analyze complex curves (like loops or vertical segments) that fail the Vertical Line Test.

Formula / Concept Box

Application	Formula	Condition
First Derivative	$\frac{dy}{dx} = \frac{dy/dt}{dx/dt}$	$dx/dt \neq 0$
Second Derivative	$\frac{d^2y}{dx^2} = \frac{\frac{d}{dt} \left( \frac{dy}{dx} \right)}{dx/dt}$	Measures concavity
Arc Length	$L = \int_{a}^{b} \sqrt{(\frac{dx}{dt})^2 + (\frac{dy}{dt})^2} \, dt$	Curve traversed once
Area Under Curve	$A = \int_{a}^{b} y(t) x'(t) \, dt$	$x(t)$ is monotonic
Surface Area ( $x$ -axis)	$S = \int_{a}^{b} 2\pi y(t) \sqrt{(x'(t))^2 + (y'(t))^2} \, dt$	$y(t) \geq 0$

Hierarchical Outline

I. Differentiation of Parametric Equations
- A. Slope of Tangent Lines: Calculated by the ratio of vertical change to horizontal change relative to $t$ .
- B. Horizontal Tangents: Occur when $dy/dt = 0$ (and $dx/dt \neq 0$ ).
- C. Vertical Tangents: Occur when $dx/dt = 0$ (and $dy/dt \neq 0$ ).
II. Integration of Parametric Equations
- A. Area: Transitioning $\int y \, dx$ into parameter space using substitution $dx = x'(t) \, dt$ .
- B. Arc Length: Derived from the Pythagorean theorem applied to infinitesimal segments $ds = \sqrt{dx^2 + dy^2}$ .
III. Surface Area of Revolution
- A. Rotation about x-axis: Uses $2\pi y$ as the circumference.
- B. Rotation about y-axis: Uses $2\pi x$ as the circumference.

Visual Anchors

Differentiation Flowchart

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Figure 1 — Mermaid diagram

Parametric Geometry

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Figure 2 — TikZ diagram

Definition-Example Pairs

Term: Arc Length Differential ( $ds$ $d s$ )
- Definition: The infinitesimal distance along a curve, $ds = \sqrt{(x'(t))^2 + (y'(t))^2} dt$ .
- Example: For a circle $x = \cos t, y = \sin t$ , $ds = \sqrt{(-\sin t)^2 + (\cos t)^2} dt = 1 \, dt$ .
Term: Second Derivative $\frac{d^2y}{dx^2}$ $\frac{d ^{2} y}{d x ^{2}}$
- Definition: The rate of change of the slope with respect to $x$ , not $t$ .
- Example: If $dy/dx = t^2$ and $x = t^3$ , the second derivative is $(2t) / (3t^2) = 2/(3t)$ .

Worked Examples

Example 1: Finding the Equation of a Tangent Line

Problem: Find the equation of the tangent line to the curve $x = t^2, y = t^3 - 3t$ at $t = 2$ .

Find Point: x(2) = 4, y(2) = 8 - 6 = 2. Point is $(4, 2)$ .
Find Derivatives: $dx/dt = 2t$ and $dy/dt = 3t^2 - 3$ .
Calculate Slope: $\frac{dy}{dx} = \frac{3t^2 - 3}{2t}$ . At $t=2$ , $m = \frac{3(4)-3}{2(2)} = \frac{9}{4}$ .
Equation: $y - 2 = \frac{9}{4}(x - 4) \implies y = \frac{9}{4}x - 7$ .

Example 2: Arc Length of a Circle

Problem: Use the arc length formula to find the circumference of a circle of radius $r$ ( $x = r\cos t, y = r\sin t$ ) for $$0 \leq t \leq 2\pi$$.

Derivatives: $x'(t)$ = -r\sin t $, $y'(t) $= r\cos t$ .
Integrand: $\sqrt{(-r\sin t)^2 + (r\cos t)^2} = \sqrt{r^2(\sin^2 t + \cos^2 t)} = r$ .
Integral: $L = \int_{0}^{2\pi} r$ \, dt = [rt]$_0^{2\pi} = 2\pi r$ .

[!TIP] Always check if a curve is traversed more than once over the given interval. If $t$ goes from 0 to $4\pi$ for a circle, the arc length formula will give $4\pi r$ , which is double the actual circumference.

Checkpoint Questions

How do you find the values of $t$ where a parametric curve has a horizontal tangent line?
Why is the second derivative of a parametric curve NOT simply $\frac{d^2y/dt^2}{d^2x/dt^2}$ ?
Set up the integral for the area under the curve $x = t + 1, y = t^2$ from $t=0$ to $t=2$ .
What visual property of the curve does the sign of $\frac{d^2y}{dx^2}$ determine?