## Summary of Calculus of Parametric Curves

### Essential Concepts

• 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

$S=2\pi {\displaystyle\int }_{a}^{b}y\left(t\right)\sqrt{{\left({x}^{\prime }\left(t\right)\right)}^{2}+{\left({y}^{\prime }\left(t\right)\right)}^{2}}dt$