Essential Concepts

Fundamentals of Parametric Equations

• Parametric equations provide a convenient way to describe a curve. A parameter can represent time or some other meaningful quantity.
• It is often possible to eliminate the parameter in a parameterized curve to obtain a function or relation describing that curve.
• There is always more than one way to parameterize a curve.
• Parametric equations can describe complicated curves that are difficult or perhaps impossible to describe using rectangular coordinates.

Calculus with Parametric Curves

• The derivative of the parametrically defined curve [latex]x=x\left(t\right)[/latex] and [latex]y=y\left(t\right)[/latex] can be calculated using the formula [latex]\frac{dy}{dx}=\frac{{y}^{\prime }\left(t\right)}{{x}^{\prime }\left(t\right)}[/latex]. 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 [latex]A={\displaystyle\int }_{{t}_{1}}^{{t}_{2}}y\left(t\right){x}^{\prime }\left(t\right)dt[/latex].
• The arc length of a parametric curve can be calculated by using the formula [latex]s={\displaystyle\int }_{{t}_{1}}^{{t}_{2}}\sqrt{{\left(\frac{dx}{dt}\right)}^{2}+{\left(\frac{dy}{dt}\right)}^{2}}dt[/latex].
• The surface area of a volume of revolution revolved around the x-axis is given by [latex]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[/latex]. If the curve is revolved around the y-axis, then the formula is [latex]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[/latex].

Key Equations

• Derivative of parametric equations
  
  [latex]\frac{dy}{dx}=\frac{\frac{dy}{dt}}{\frac{dx}{dt}}=\frac{{y}^{\prime }\left(t\right)}{{x}^{\prime }\left(t\right)}[/latex]
• Second-order derivative of parametric equations
  
  [latex]\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}}[/latex]
• Area under a parametric curve
  
  [latex]A={\displaystyle\int }_{a}^{b}y\left(t\right){x}^{\prime }\left(t\right)dt[/latex]
• Arc length of a parametric curve
  
  [latex]s={\displaystyle\int }_{{t}_{1}}^{{t}_{2}}\sqrt{{\left(\frac{dx}{dt}\right)}^{2}+{\left(\frac{dy}{dt}\right)}^{2}}dt[/latex]
• Surface area generated by a parametric curve
  
  [latex]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[/latex]

Glossary

cusp

a pointed end or part where two curves meet

cycloid

the curve traced by a point on the rim of a circular wheel as the wheel rolls along a straight line without slippage

orientation

the direction that a point moves on a graph as the parameter increases

parameter

an independent variable that both x and y depend on in a parametric curve; usually represented by the variable t

parametric curve

the graph of the parametric equations [latex]x\left(t\right)[/latex] and [latex]y\left(t\right)[/latex] over an interval [latex]a\le t\le b[/latex] combined with the equations

parametric equations

the equations [latex]x=x\left(t\right)[/latex] and [latex]y=y\left(t\right)[/latex] that define a parametric curve

parameterization of a curve

rewriting the equation of a curve defined by a function [latex]y=f\left(x\right)[/latex] as parametric equations