Calculus with Parametric Equations

Unit 10.2 of Math 152

Derivatives with Parametric Curves

• Much like differential equations we can think about this as the derivative version of it

  • Definition

    • $\frac{dy}{dx}=\frac{\frac{dy}{dt}}{\frac{dx}{dt}}$ where $\frac{dx}{dt}=/=0$ very simple :)

Integrals with Parametric Curves

• Much like derivatives with parametric curves we have to treat it as functions for each input

  • Definition

    • $A={\int }_a^{b}ydx⟹{\int }_{\alpha }^{\beta }g(t)f^{\prime }(t)dt$

Arc Length with Parametric Curves

• Knowing what we know about Arc Length

  • The equation $L={\int }_a^b\sqrt{1+{\frac{dy}{dx}}^2}dx$

    • this describes only normal Cartesian functions however if we want it for Parametric functions we must
      • then we can us the derivatives with parametric curves equation and sub it for $\frac{dy}{dx}$
        • $⟹L={\int }_{\alpha }^{\beta }\sqrt{(\frac{dx}{dt}{)}^2+{\left(\frac{dy}{dt}\right)}^2}dt$

Surface Area

• very simple as well :) same thing about Arc Length

• Definition

  • $S={\int }_{\alpha }^{\beta }2\pi y\sqrt{{\left(\frac{dx}{dt}\right)}^2+{\left(\frac{dy}{dt}\right)}^2}dt$