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Understanding Polar Areas and Curves in AP Calculus

This guide elaborates on essential concepts in AP Calculus related to polar curves, focusing on calculating areas associated with cardioids and rose curves. Topics include determining the diameter of curves, using equations to find points of intersection, and applying trigonometric substitution for polar regions. Clear examples demonstrate how to find areas bounded by specific curves, including regions of intersection. The principles discussed are vital for mastering polar coordinates and areas, and preparing for calculus exams.

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Understanding Polar Areas and Curves in AP Calculus

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  1. 5047 Polar Area BC Bonus Polar Functions - Area BC CALCULUS AP Calculus

  2. REVIEW Polar Curves: a gives the Diameter (or length of a Leaf) Circles b < a with loop b = a Cardioid b > a no loop Limaζon n(odd) = n petals n(even) = 2n petals Rose

  3. REVIEW Intersections: • Set the Equations equal and solve. • Check the pole independently in each curve.

  4. REVIEW Intersections: • Set the Equations equal and solve. • Check the pole independently in each curve.

  5. REVIEW Intersections: • Set the Equations equal and solve. • Check the pole independently in each curve.

  6. Lemma: Trig substitution

  7. Area of a Circular Sector REM: Calculus works with RADIANS.

  8. Area ofa Polar Region REM:

  9. Illustration: Find the area of the region bounded by the cardioid.

  10. Example: Find the area of the region bounded by the one leaf of the rose.

  11. Example: Region of Intersection Find the area of the region in the intersection between the curves.

  12. Example: Region between curves Find the area of the region inside the cardioid and outside the circle.

  13. LAST UPDATE • 03/19/12 • Assignment : p. 558 # 43 – 59 odd

  14. Area of a Circular Sector REM: Calculus works with RADIANS. -

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