Chap10_Sec4 - 10 PARAMETRIC EQUATIONS AND POLAR COORDINATES...

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PARAMETRIC EQUATIONS PARAMETRIC EQUATIONS AND POLAR COORDINATES AND POLAR COORDINATES 10
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10.4 Areas and Lengths in Polar Coordinates In this section, we will: Develop the formula for the area of a region whose boundary is given by a polar equation. PARAMETRIC EQUATIONS & POLAR COORDINATES
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AREAS IN POLAR COORDINATES We need to use the formula for the area of a sector of a circle A = ½ r 2 θ where: r is the radius. θ is the radian measure of the central angle. Formula 1
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AREAS IN POLAR COORDINATES Formula 1 follows from the fact that the area of a sector is proportional to its central angle: A = ( θ/ 2 π ) πr 2 = ½ r 2 θ
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AREAS IN POLAR COORDINATES Let R be the region bounded by the polar curve r = f ( θ ) and by the rays θ = a and θ = b, where: f is a positive continuous function. 0 < b – a ≤ 2 π
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AREAS IN POLAR COORDINATES We divide the interval [ a, b ] into subintervals with endpoints θ 0 , θ 1 , θ 2 , …, θ n , and equal width ∆θ . Then, the rays θ = θ i divide R   into smaller regions with central angle ∆θ = θ i θ i –1 .
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AREAS IN POLAR COORDINATES If we choose θ i * in the i th subinterval [ θ i –1 , θ i ] then the area ∆A i of the i th region is the area of the sector of a circle with central angle ∆θ and radius f ( θ* ).
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AREAS IN POLAR COORDINATES Thus, from Formula 1, we have: A i ½[ f ( θ i *)] 2 θ So, an approximation to the total area A of R   is: * 2 1 2 1 [ ( )] n i i A f θ θ = Formula 2
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AREAS IN POLAR COORDINATES It appears that the approximation in Formula 2 improves as n .
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AREAS IN POLAR COORDINATES However, the sums in Formula 2 are Riemann sums for the function g ( θ ) = ½[ f ( θ )] 2 . So, * 2 2 1 1 2 2 1 lim [ ( )] [ ( )] n b i a n i f f d θ θ θ θ →∞ = =
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AREAS IN POLAR COORDINATES Therefore, it appears plausible—and can, in fact, be proved—that the formula for the area A of the polar region R is: 2 1 2 [ ( )] b a A f d θ θ = Formula 3
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AREAS IN POLAR COORDINATES Formula 3 is often written as with the understanding that r = f ( θ ).
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