Key Equations

Key Equations

Derivative of parametric equations	$\frac{dy}{dx}=\frac{dy\text{/}dt}{dx\text{/}dt}=\frac{y^{\prime }\left(t\right)}{x^{\prime }\left(t\right)}$
Second-order derivative of parametric equations	$\frac{d^{2}y}{d{x}^2}=\frac{d}{dx}\left(\frac{dy}{dx}\right)=\frac{\left(d\text{/}dt\right)\left(dy\text{/}dx\right)}{dx\text{/}dt}$
Area under a parametric curve	$A={\displaystyle {\int }_a^{b}y\left(t\right)x^{\prime }\left(t\right)dt}$
Arc length of a parametric curve	$s={\displaystyle {\int }_{t_1}^{t_2}\sqrt{{\left(\frac{dx}{dt}\right)}^2+{\left(\frac{dy}{dt}\right)}^2}dt}$
Surface area generated by a parametric curve	$S=2\pi {\displaystyle {\int }_a^{b}y\left(t\right)\sqrt{{\left(x^{\prime }\left(t\right)\right)}^2+{\left(y^{\prime }\left(t\right)\right)}^2}dt}$

Area of a region bounded by a polar curve	$A=\frac{1}{2}{\displaystyle {\int }_{\alpha }^{\beta }{\left[f\left(\theta \right)\right]}^{2}d\theta }=\frac{1}{2}{\displaystyle {\int }_{\alpha }^{\beta }{r}^{2}d\theta }$
Arc length of a polar curve	$L={\displaystyle {\int }_{\alpha }^{\beta }\sqrt{{\left[f\left(\theta \right)\right]}^2+{\left[f^{\prime }\left(\theta \right)\right]}^2}}d\theta ={\displaystyle {\int }_{\alpha }^{\beta }\sqrt{r^2+{\left(\frac{dr}{d\theta }\right)}^2}d\theta }$