Summary of Calculus of Parametric Curves

Essential Concepts

  • 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]