Parametric Equations

Parametric Curves

Let $x$ and $y$ be functions in $t$ on an interval $ℐ$

x = x (t), y = y (t)

The set of points

(x, y) = (x (t), y (t))

is called a parametric curve. The variable $t$ is called parameter. The functions are parametric equations for the curve.

Derivative

Let $x = x (t)$ and $y = y (t)$ . If $x^{'} (t)$ and $y^{'} (t)$ exist and $x^{'} (t) \neq 0$ then

\frac{ⅆ y}{ⅆ x} = \frac{\frac{ⅆ y}{ⅆ t}}{\frac{ⅆ x}{ⅆ t}}

For the second derivative

\frac{ⅆ \frac{ⅆ y}{ⅆ x}}{ⅆ x} = \frac{\frac{ⅆ}{ⅆ t} (\frac{ⅆ y}{ⅆ x})}{\frac{ⅆ x}{ⅆ t}}

Area

Let $x = x (t)$ and $y = y (t)$ for $t \in [𝒶, 𝒷]$ define a non-intersecting curve. Assume $x (t)$ is differentiable. Then the area $𝒜$ under the curve is

𝒜 = \int_{𝒶}^{𝒷} y (t) ⅆ x (t) = \int_{𝒶}^{𝒷} y (t) x^{'} (t) ⅆ t

Area approximation

Arc Length

Let $x = x (t)$ and $y = y (t)$ for $t \in [𝒶, 𝒷]$ be differentiable. Then the arc length $ℒ$ of the corresponding parametric curve is

ℒ = \int_{𝒶}^{𝒷} \sqrt{{(x^{'} (t))}^{2} + {(y^{'} (t))}^{2}} ⅆ t

Pythagorean approximation

Surface of revolution

Let $x = x (t)$ and $y = y (t)$ for $t \in [𝒶, 𝒷]$ be differentiable. Then the surface area $𝒮$ obtained by revolving the curve around the $x$ -axis is