Section 8'3 Area and Arc Length in Polar Coordinates

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Section 8.3Area and Arc Length in Polar
Coordinates
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Review of polar coordinates

• Any point in the place can be represented as
  rectangular coordinates or polar coordinates
• Where rectangular coordinates require the
  horizontal and vertical distance from the origin,
  polar coordinates require the distance from the
  origin and the angle between the ray the point
  lies on and the positive side of the horizontal
  axis.
• Lets take a look

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• Suppose P is the indicated point in the xy-plane
• Then
• What are the rectangular coordinates given by
  (5,p/6)?
• What are the polar coordinates given by (-4, 5)

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Graphs of Equations in Polar Coordinates

• Some graphs are easier to represent in polar
  coordinates
• What does the graph r 2 look like?
• What does the graph r ? look like?
• Lets take a look at these and some other graphs
  using Maple
• Note You can graph these in your graphing
  calculator as well by switching the mode to polar
• Now lets talk about calculating the area
  enclosed by a polar graph

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• Recall the area of a sector
• Imagine the area of our polar graph being broken
  up into infinitely small sectors
• Lets find the area of a four leaved rose

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• Lets find the area inside r 3cos? and outside
  r 1 cos?
• Here is a plot

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Arc Length in Polar Coordinates

• Recall our Arc Length formula
• We will use this to derive the Arc Length formula
  for polar coordinates
• And we get