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HH_Sec_8_3 - r = 4 sin θ ≤ θ ≤ π in Figure 1 θ 2-2...

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(9/30/08) Math 10B. Lecture Examples. Section 8.3. Area and arc length in polar coordinates Example 1 Sketch the curve in an xy -plane with the polar equation r = 1 + cos θ, 0 θ 2 π . (The curve is called a cardioid because of its heart-like shape.) Answer: Figures A1a, A1b, and A1c θ r 1 2 π 2 π r = 1 + cos θ 1 2 π 3 2 π x 1 y - 1 1 r θ x 1 y - 1 1 r = 1 + cos θ r = 1 + cos θ in an -plane The cardioid for 0 t 1 2 π The entire cardioid Figure A1a Figure A1b Figure A1c Example 2 Find the area of the region bounded by the cardioid from Example 1 with polar equation r = 1 + cos θ, 0 θ 2 π . (Use the identity cos 2 θ = 1 2 [ 1 + cos ( 2 θ )] in the integration.) Answer: [Area] = 3 2 π Example 3 Use polar coordinates to find the area bounded by the circle
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Unformatted text preview: r = 4 sin θ, ≤ θ ≤ π in Figure 1. θ 2-2 r 2 r = 4 sin θ FIGURE 1 Interactive Examples Work the following Interactive Examples on Shenk’s web page, http//www.math.ucsd.edu/˜ashenk/: ‡ Section 11.3: Examples 1–5 † Lecture notes to accompany Section 8.3 of Calculus by Hughes-Hallett et al. ‡ The chapter and section numbers on Shenk’s web site refer to his calculus manuscript and not to the chapters and sections of the textbook for the course. 1...
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  • Summer '10
  • reed
  • Coordinate system, Spherical coordinate system, Polar coordinate system, Conic section, Cylindrical coordinate system

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