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Area in Polar Coordinates

Area in Polar Coordinates. Lesson 10.10. 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. Area of a Sector of a Region. Consider a region bounded by r = f( θ )

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Area in Polar Coordinates

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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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