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Chapter 10: Parametric Equations and Polar Coordinates

Learning module LM 10.1: Parametrized Curves:

Learning module LM 10.2: Calculus with Parametrized Curves:

      Slope and area
      Arc 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 Space


Chapter 13: Vector Functions


Chapter 14: Partial Derivatives


Chapter 15: Multiple Integrals



Summary and simplification

Summary and Simplification
Our first set of results applies to all parametrized curves:

If we have a parametrized curve, where we know x(t) and y(t), then

    The slope of the tangent line at time t is dy/dtdx/dt.

    The area under the curve from time t1 to t2 is t2t1y(t)dx(t)dtdt.

    The distance traveled along the curve from time t1 to time t2 is t2t1(dxdt)2+(dydt)2dt

    The area of the surface of revolution, obtained by rotating around the x-axis, is t2t12πy(t)(dxdt)2+(dydt)2dt


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 slope is dy/dtdx/dt=f(t)1=f(x).

    The area under the curve is t2t1f(t)dt=x2x1f(x)dx.

    The arc length is t2t11+(f(t))2dt=x2x11+(f(x))2dx.

    The surface area is t2t12πf(t)1+(f(t))2dt=x2x12πf(x)1+(f(x))2dx.


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.