Summary of Calculus of Parametric Curves

The derivative of the parametrically defined curve $x=x\left(t\right)$ and $y=y\left(t\right)$ can be calculated using the formula $\frac{dy}{dx}=\frac{{y}^{\prime }\left(t\right)}{{x}^{\prime }\left(t\right)}$ . Using the derivative, we can find the equation of a tangent line to a parametric curve.
The area between a parametric curve and the x-axis can be determined by using the formula $A={\displaystyle\int }_{{t}_{1}}^{{t}_{2}}y\left(t\right){x}^{\prime }\left(t\right)dt$ .
The arc length of a parametric curve can be calculated by using the formula $s={\displaystyle\int }_{{t}_{1}}^{{t}_{2}}\sqrt{{\left(\frac{dx}{dt}\right)}^{2}+{\left(\frac{dy}{dt}\right)}^{2}}dt$ .
The surface area of a volume of revolution revolved around the x-axis is given by $S=2\pi {\displaystyle\int }_{a}^{b}y\left(t\right)\sqrt{{\left({x}^{\prime }\left(t\right)\right)}^{2}+{\left({y}^{\prime }\left(t\right)\right)}^{2}}dt$ . If the curve is revolved around the y-axis, then the formula is $S=2\pi {\displaystyle\int }_{a}^{b}x\left(t\right)\sqrt{{\left({x}^{\prime }\left(t\right)\right)}^{2}+{\left({y}^{\prime }\left(t\right)\right)}^{2}}dt$ .

Key Equations

Derivative of parametric equations
$\frac{dy}{dx}=\frac{\frac{dy}{dt}}{\frac{dx}{dt}}=\frac{{y}^{\prime }\left(t\right)}{{x}^{\prime }\left(t\right)}$
Second-order derivative of parametric equations
$\frac{{d}^{2}y}{d{x}^{2}}=\frac{d}{dx}\left(\frac{dy}{dx}\right)=\frac{\left(\frac{d}{dt}\right)\left(\frac{dy}{dx}\right)}{\frac{dx}{dt}}$
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
$S=2\pi {\displaystyle\int }_{a}^{b}y\left(t\right)\sqrt{{\left({x}^{\prime }\left(t\right)\right)}^{2}+{\left({y}^{\prime }\left(t\right)\right)}^{2}}dt$