Lesson 20: Integration in Polar Coordinates

Section 12.4Integration in Polar CoordinatesMath 21aMarch 31, 2008Announcements◮ Office hours Tuesday, Wednesday 2–4pm SC 323◮ Problem Sessions: Mon, 8:30; Thur, 7:30; SC 103bImage: Flickr user duncanAnnouncements◮ Office hours Tuesday, Wednesday 2–4pm SC 323◮ Problem Sessions: Mon, 8:30; Thur, 7:30; SC 103bOutlineLast TimePolar geometrySectorsPolar RectanglesIntegration over polar regionsFact of the dayWorksheetNext TimeFactIf D is a region of Type I:D = { (x, y) | a ≤ x ≤ b, g (x) ≤ y ≤ g (x) }12Then for any “mostly” continuous function f∫∫∫ ∫bg (x)2f(x, y) dA =f(x, y) dy dxDag (x)1FactIf D is a region of Type II:D = { (x, y) | c ≤ y ≤ d, h1(y) ≤ x ≤ h2(y) }Then for any “mostly” continuous function f∫∫∫ ∫dh2(y)f(x, y) dA =f(x, y) dx dyDch1(y)This timeCan polar coordinates help in computing integrals?OutlineLast TimePolar geometrySectorsPolar RectanglesIntegration over polar regionsFact of the dayWorksheetNext TimeArc length and sector areas = rθs1A =r2θ2(if θ is in radians)θrArea of a polar rectangleθ21¯r∆A =r2r22 2 (θ2 − θ1) − 12 1 (θ2 − θ1)()1=r2 − r2 (θ2 − θ1)2211=(r2 + r1) (r2 − r1) (θ2 − θ1)2= ¯r∆r ∆θ∆θθ1r1r2OutlineLast TimePolar geometrySectorsPolar RectanglesIntegration over polar regionsFact of the dayWorksheetNext TimeDocument Outline

• Announcements
• Last Time
• Polar geometry
  • Sectors
  • Polar Rectangles
• Integration over polar regions
  • Fact of the day
• Worksheet
• Next Time