M408M Learning Module Pages
Main page Chapter 10: Parametric Equations and Polar CoordinatesLearning module LM 10.1: Parametrized Curves:Learning module LM 10.2: Calculus with Parametrized Curves:Slope and areaArc length and surface area Summary and simplification Higher Derivatives Learning module LM 10.3: Polar Coordinates:Learning module LM 10.4: Areas and Lengths of Polar Curves:Learning module LM 10.5: Conic Sections:Learning module LM 10.6: Conic Sections in Polar Coordinates:Chapter 12: Vectors and the Geometry of SpaceChapter 13: Vector FunctionsChapter 14: Partial DerivativesChapter 15: Multiple Integrals |
Summary and simplification
If we happen to have a graph y=f(x) then all of these formulas simplify, since we can take the parametrization x(t)=t, y(t)=f(t). In that case dxdt=1 and dydt=f′(t)=f′(x), so
The first two are our usual formulas for slopes and areas. The third and fourth may (or may not) be familiar to you from applications of integration. |