Lengths and Surface Areas of Polar Curves Section 10.6b.

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Presentation transcript:

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 Now substitute back into the original formula…

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

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

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 :

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 :

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. traces the entire graph on Check p.565 for a graph and diagram… First, evaluate the inside of the square root:

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…

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…