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Lengths and Surface Areas of Polar Curves

Lengths and Surface Areas of Polar Curves. Section 10.6b. Length of a Polar Curve. Start with a parametrization of , :. Length of a parametric curve (from Section 10.1):. Inside the square root:. Length of a Polar Curve.

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Lengths and Surface Areas of Polar Curves

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  1. Lengths and Surface Areas of Polar Curves Section 10.6b

  2. Length of a Polar Curve Start with a parametrization of , : Length of a parametric curve (from Section 10.1): Inside the square root:

  3. Length of a Polar Curve Now substitute back into the original formula…

  4. Length of a Polar Curve If has a continuous first derivative for and if the point traces the curve exactly once as runs from to , then the length of the curve is

  5. Length of a Polar Curve Find the length of the given cardioid. Check the graph for the angle interval. (Using NINT)

  6. Area of a Surface of Revolution If has a continuous first derivative for and if the point traces the curve exactly exactly once as runs from to , then the areas of the surfaces generated by revolving the curve about the x- and y-axes are given by the following formulas: Revolution about the x-axis :

  7. Area of a Surface of Revolution If has a continuous first derivative for and if the point traces the curve exactly exactly once as runs from to , then the areas of the surfaces generated by revolving the curve about the x- and y-axes are given by the following formulas: Revolution about the y-axis :

  8. Area of a Surface of Revolution Find the area of the surface generated by revolving the right- hand loop of the given lemniscate curve about the y-axis. Check p.565 for a graph and diagram… traces the entire graph on First, evaluate the inside of the square root:

  9. Area of a Surface of Revolution Find the area of the surface generated by revolving the right- hand loop of the given lemniscate curve about the y-axis. Check p.565 for a graph and diagram… Now substitute back into the original formula…

  10. Area of a Surface of Revolution Find the area of the surface generated by revolving the right- hand loop of the given lemniscate curve about the y-axis. Check p.565 for a graph and diagram…

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