11'2 Calculus with Parameterized Curves

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11.2 Calculus with Parameterized Curves

• Math 6B
• Calculus II

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Tangents

• To find the slope of the tangent dy/dx from the
  parametric equations x f (t) and y
  g (t), let us use the chain rule of dy/dt

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Tangents

• We can get dy/dx by itself and therefore get the
  slope of the tangent line.

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Second Derivative
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Area

• Area under a curve is defined by
• rewriting the integral we get

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Arc Length

• Remember from Chapter 9, the arc length formula
  is
• Once again rewriting the integral we get

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Arc Length

• Theorem If a smooth curve, x f (t), y
  g (t), , is traversed exactly
  once as t increases from a to b, the curves
  length is

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Surface Area

• If the curve given by the parametric equations x
  f (t), y g (t), is
  rotated about the x axis, where f and g are
  continuous and , then the area of
  the resulting surface is given by

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Surface Area