Key Equations

Derivative of parametric equations	$\frac{d y}{d x} = \frac{d y / d t}{d x / d t} = \frac{y^{'} (t)}{x^{'} (t)}$
Second-order derivative of parametric equations	$\frac{d^{2} y}{d x^{2}} = \frac{d}{d x} (\frac{d y}{d x}) = \frac{(d / d t) (d y / d x)}{d x / d t}$
Area under a parametric curve	$A = \int_{a}^{b} y (t) x^{'} (t) d t$
Arc length of a parametric curve	$s = \int_{t_{1}}^{t_{2}} \sqrt{{(\frac{d x}{d t})}^{2} + {(\frac{d y}{d t})}^{2}} d t$
Surface area generated by a parametric curve	$S = 2 π \int_{a}^{b} y (t) \sqrt{{(x^{'} (t))}^{2} + {(y^{'} (t))}^{2}} d t$

Area of a region bounded by a polar curve	$A = \frac{1}{2} \int_{α}^{β} {[f (θ)]}^{2} d θ = \frac{1}{2} \int_{α}^{β} r^{2} d θ$
Arc length of a polar curve	$L = \int_{α}^{β} \sqrt{{[f (θ)]}^{2} + {[f^{'} (θ)]}^{2}} d θ = \int_{α}^{β} \sqrt{r^{2} + {(\frac{d r}{d θ})}^{2}} d θ$