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Calculating Areas in Polar Coordinates - Lesson on Sectors and Regions

Learn how to find areas of sectors and regions in polar coordinates using integrals. Explore different functions inside and outside circles and ellipses. Practice with exercises provided.

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Calculating Areas in Polar Coordinates - Lesson on Sectors and Regions

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  1. Area in Polar Coordinates Lesson 10.10

  2. Area of a Sector of a Circle • Given a circle with radius = r • Sector of the circle with angle = θ • The area of the sector given by θ r

  3. Area of a Sector of a Region • Consider a region bounded by r = f(θ) • A small portion (a sector with angle dθ) has area β • dθ α •

  4. Area of a Sector of a Region • We use an integral to sum the small pie slices β • r = f(θ) α •

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

  6. The ellipse is traced out by 0 < θ < 2π Find the Area • Given r = 4 + sin θ • Find the area of the region enclosed by the ellipse dθ

  7. Areas of Portions of a Region • Given r = 4 sin θ and rays θ = 0, θ = π/3 The angle of the rays specifies the limits of the integration

  8. Area of a Single Loop • Consider r = sin 6θ • Note 12 petals • θ goes from 0 to 2π • One loop goes from0 to π/6

  9. Area Of Intersection • Note the area that is inside r = 2 sin θand outside r = 1 • Find intersections • Consider sector for a dθ • Must subtract two sectors dθ

  10. Assignment • Lesson 10.10A • Page 459 • Exercises 1 – 19 odd • Lesson 10.10B • Page 459 • Exercises 21 – 27 odd

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