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Consider the region bounded by the curve r = f ( θ ) and the two rays

Integration in polar coordinates involves finding not the area underneath a curve but, rather, the area of a sector bounded by a curve . . Consider the region bounded by the curve r = f ( θ ) and the two rays.

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Consider the region bounded by the curve r = f ( θ ) and the two rays

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  1. Integration in polar coordinates involves finding not the area underneath a curve but, rather, the area of a sector bounded by a curve. Consider the region bounded by the curve r = f (θ) and the two rays To derive a formula for the area, divide the region into N narrow sectors of angle corresponding to a partition of the interval

  2. Recall that a circular sector of angle Δθand radius rhas If Δθ is small, the jth narrow sector is nearly a circular sector of radius rj = f (θj),so its area is The total area is approximated by the sum:

  3. This is a Riemann sum for the integral If f (θ) is continuous, then the sum approaches the integral as and we obtain the following formula. THEOREM 1 Area in Polar Coordinates If f (θ) is a continuous function, then the area bounded by a curve in polar form r= f (θ) and the rays θ =

  4. We know that r = R defines a circle of radius R. By THM 1, the area is equal to THEOREM 1 Area in Polar Coordinates If f (θ) is a continuous function, then the area bounded by a curve in polar form r= f (θ) and the rays θ =

  5. Use Theorem 1 to compute the area of the right semicircle with equation r = 4 sin θ. The equation r= 4 sin θdefines a circle of radius 2 tangent to the x-axis at the origin. The right semicircle is “swept out” as θ varies from 0 to By THM 1, the area of the right semicircle is

  6. Sketch r = sin 3θand compute the area of one “petal.” r varies from 0 to 1 and back to 0 as θ varies from 0 to r varies from 0 to -1 and back to 0 as θ varies from r varies from 0 to 1 and back to 0 as θ varies from

  7. The area between two polar curves r = f1(θ) and r = f2(θ) with f2(θ) ≥ f1(θ), for is equal to Area Between Two Curves Find the area of the region inside the circle r= 2 cosθ but outside the circle r = 1. The two circles intersect at the points where (r, 2 cosθ) = (r, 1) or in other words, when 2 cosθ = 1. Region (I) is the difference of regions (II) and (III).

  8. We close this section by deriving a formula for arc length in polar coordinates. Observe that a polar curve r = f (θ) has a parametrization with θ as a parameter:

  9. Find the total length of the circle r = 2acosθ for a > 0. f (θ) = 2acosθ Note that the upper limit of integration is π rather than 2π because the entire circle is traced out as θ varies from 0 to π.

  10. To find the slope of a polar curve r = f (θ),remember that the curve is in the x-y plane, and so the slope is Since x = rcosθand y = r sin θ, we use the chain rule.

  11. Find an equation of the line tangent to the polar curve r= sin 2θwhen

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