Parametric Equations x(t), y(t)

Curve defined by: x = f(t), y = g(t), t ∈ [a, b]

dy/dx = (dy/dt)/(dx/dt)

$\frac{dy}{dx} = \frac{dy/dt}{dx/dt} = \frac{g'(t)}{f'(t)}$

d2y/dx2 = d/dt (dy/dx) / (dx/dt)

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

Arc Length = ∫ab(dx/dt)2+(dy/dt)2dt\int_a^b \sqrt{(dx/dt)^2 + (dy/dt)^2} dt∫ab​(dx/dt)2+(dy/dt)2​dt

$L = \int_a^b \sqrt{(f'(t))^2 + (g'(t))^2}\,dt$

Vector-Valued Function r(t) = <x(t), y(t)>

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

Integral of r(t) = <∫x(t)dt,∫y(t)dt\int x(t)dt, \int y(t)dt∫x(t)dt,∫y(t)dt>

$\int \vec{r}(t)dt = \langle \int x(t)dt, \int y(t)dt \rangle$

• $\vec{v}(t) = \vec{r}'(t)$
• $\vec{a}(t) = \vec{r}''(t)$
• Speed = $|\vec{v}(t)|$

Polar Coordinates (r, θ)

$x = r\cos\theta, \quad y = r\sin\theta$

$\frac{dy}{dx} = \frac{r'\sin\theta + r\cos\theta}{r'\cos\theta - r\sin\theta}$

where $r' = \frac{dr}{d\theta}$

Area = (1/2) ∫αβ[r(θ)]2dθ\int_\alpha^\beta [r(\theta)]^2 d\theta∫αβ​[r(θ)]2dθ

$A = \frac{1}{2}\int_\alpha^\beta [r(\theta)]^2 d\theta$

Area = (1/2) ∫αβ([R(θ)]2−[r(θ)]2)dθ\int_\alpha^\beta ([R(\theta)]^2 - [r(\theta)]^2) d\theta∫αβ​([R(θ)]2−[r(θ)]2)dθ

$A = \frac{1}{2}\int_\alpha^\beta ([R(\theta)]^2 - [r(\theta)]^2)d\theta$

where R(θ) ≥ r(θ) on [alpha, β].

Arc Length = ∫αβr(θ)2+(dr/dθ)2dθ\int_\alpha^\beta \sqrt{r(\theta)^2 + (dr/d\theta)^2} d\theta∫αβ​r(θ)2+(dr/dθ)2​dθ

$L = \int_\alpha^\beta \sqrt{r^2 + \left(\frac{dr}{d\theta}\right)^2}\,d\theta$