Area and Arc Length in Polar Coordinates

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Area and Arc Length in Polar Coordinates

• Lesson 10.5

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Area of a Sector of a Circle

• Given a circle with radius r
• Sector of the circle with angle ?
• The area of the sector given by

?
r
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Area of a Sector of a Region

• Consider a region bounded by r f(?)
• A small portion (a sector with angle d?) has area

ß

d?
a

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Area of a Sector of a Region

• We use an integral to sum the small pie slices

ß

r f(?)
a

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Guidelines

• Use the calculator to graph the region
• Find smallest value ? a, and largest value ?
  b for the points (r, ?) in the region
• Sketch a typical circular sector
• Label central angle d?
• Express the area of the sector as
• Integrate the expression over the limits from a
  to b

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Find the Area

• Given r 4 sin ?
• Find the area of the region enclosed by the
  ellipse

d?
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Areas of Portions of a Region

• Given r 4 sin ? and rays ? 0, ? p/3

The angle of the rays specifies the limits of the
integration
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Area of a Single Loop

• Consider r sin 6?
• Note 12 petals
• ? goes from 0 to 2p
• One loop goes from0 to p/6

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Intersections of Polar Equations

• To find area of intersecting regions
• Need to know where the graphs intersect

• r 1
• r 2 cos ?

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Strategies

• r 1
• r 2 cos ?

• Use substitution
• Let r 1 in the second equation
• Solve for ?
• Let _at_n1 0, result is

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A Sneaky Problem

• Consider r sin ? and r cos ?
• What is simultaneoussolution?
• Where sin ? cos ? that is
• Problem the intersection at the pole does not
  show up using this strategy
• You must inspect the graph

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Hints

• Graph the curves on your calculator
• Observe the number of intersections
• Zoom in as needed
• Do a simultaneous solution to the two equations
• Check results against observed points of
  intersection
• Discard duplicates
• Note intersection at the pole that simultaneous
  solutions may not have given

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Area Of Intersection

• Note the area that is inside r 2 sin ?and
  outside r 1
• Find intersections
• Consider sector for a d?
• Must subtract two sectors

d?
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Arc Length

• Given a curve in polar form r f (?)
• Must have continuous first derivative on interval
• Curve must be traced exactly once for a ? b
• Arc length is

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Try it Out!

• Given polar function
• What is the arc length from ? 0 to ? 4
• Find dr/d?
• What is the integral and its evaluation

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Assignment

• Lesson 10.5
• Page 745
• Exercises 1 37 EOO