Section 1.2 - Calculus with Parametric Curves

Section 1.2 - Calculus with Parametric Curves

Introduction to Parametric Curves in Calculus

Parametric equations allow us to represent curves by expressing both x and y as functions of a third variable, typically t. Calculus techniques can be applied to these curves to analyze tangents, arc length, and surface area, which are essential for understanding the geometry and applications of parametric curves.

Tangents to Parametric Curves

To find the tangent line at a point on a parametric curve, we use the derivatives of the parametric equations. Suppose x = f(t) and y = g(t) are differentiable functions of t. The slope of the tangent is given by the chain rule:

• Key Formula: $\frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}}$ (provided $\frac{dx}{dt} \neq 0$)

• Horizontal Tangent: Occurs when $\frac{dy}{dt} = 0$ and $\frac{dx}{dt} \neq 0$.

• Vertical Tangent: Occurs when $\frac{dx}{dt} = 0$ and $\frac{dy}{dt} \neq 0$.

Example: For a cycloid defined by $x = r(\theta - \sin\theta)$, $y = r(1 - \cos\theta)$, the points where the tangent is horizontal or vertical can be found by setting the respective derivatives to zero.

Area Under a Parametric Curve

The area under a parametric curve can be computed using integration. If the curve is given by $x = f(t)$ and $y = g(t)$, the area A under the curve from $t = a$ to $t = b$ is:

• Key Formula: $A = \int_a^b y \frac{dx}{dt} dt$

Example: Find the area under one arch of the cycloid $x = r(\theta - \sin\theta)$, $y = r(1 - \cos\theta)$ by integrating $y \frac{dx}{d\theta}$ over the appropriate interval.

Arc Length of Parametric Curves

The length of a curve described by parametric equations $x = f(t)$ and $y = g(t)$, for $t$ in $[a, b]$, is given by:

• Key Formula: $L = \int_a^b \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2} dt$

Theorem: If $C$ is described by $x = f(t)$, $y = g(t)$, $a \leq t \leq b$, and $f', g'$ are continuous, then the length of $C$ is:

• $L = \int_a^b \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2} dt$

Example: Find the length of one arch of the cycloid using the above formula.

Surface Area Generated by Rotating Parametric Curves

When a parametric curve is rotated about an axis, the surface area can be calculated using integration. For a curve $x = f(t)$, $y = g(t)$:

• Rotating about the x-axis: $S = \int_a^b 2\pi y \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2} dt$

• Rotating about the y-axis: $S = \int_a^b 2\pi x \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2} dt$

Example: Show that the surface area of a sphere of radius $r$ is $4\pi r^2$ by rotating the semicircle $x = r\cos t$, $y = r\sin t$, $0 \leq t \leq \pi$ about the x-axis.

Summary Table: Key Formulas for Parametric Curves

Concept	Formula	Conditions
Slope of Tangent	$\frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}}$	$\frac{dx}{dt} \neq 0$
Area Under Curve	$A=\int_{a}^{b}yx^{\prime}dt$	Curve given by $x = f(t)$, $y = g(t)$
Arc Length	$L=\int_{a}^{b}\sqrt{x^{\prime2}+y^{\prime2}}dt$	$f', g'$ continuous on $[a, b]$
Surface Area (x-axis)	$S = \int_a^b 2\pi y \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2} dt$	Rotation about x-axis
Surface Area (y-axis)	$S = \int_a^b 2\pi x \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2} dt$	Rotation about y-axis

Additional info:

• Parametric equations are especially useful for curves that cannot be represented as functions in the form $y = f(x)$.

• Applications include physics (motion along a path), engineering (design of gears and cams), and computer graphics (drawing complex shapes).