Chapter 1 Topics in Analytic Geometry

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MA112 Section 750001: Prepared by Dr.Archara Pacheenburawana 28Area in Polar Coordinatesr = f(θ)θ = β0Rθ = αArea Problem in Polar CoordinatesSuppose that α and β are angles that satisfy the conditionα < β ≤ α+2πand suppose that f(θ) is continuous and nonnegative for α ≤ θ ≤ β. Find the area of theregion R enclosed by the polar curve r = f(θ) and the rays θ = α and θ = β.Area in Polar CoordinatesIf α and β are angles that satisfy the conditionα < β ≤ α+2πand if f(θ) is continuous and either nonnegative or nonpositive for α ≤ θ ≤ β, then thearea A of the region R enclosed by the polar curve r = f(θ) (α ≤ θ ≤ β) and the linesθ = α and θ = β isA =∫ βα∫ β12 [f(θ)]2 1dθ =2 r2 dθ (2.9)αThe hardest part of applying (2.9) is determining the limits of integration. This can bedone as follows:Step 1 Sketch the region R whose area is to be determined.Step 2 Draw an arbitrary “radial line” from the pole to the boundary curve r = f(θ).Step 3 Ark, “Over what intervals must θ vary in order for the radial line to sweep out theregion R?”Step 4 The answer in Step 3 will determine the lower and upper limits of integration.
MA112 Section 750001: Prepared by Dr.Archara Pacheenburawana 29Example 2.23 Find the area of the region in the first quadrant that is within the cardioidr = 1−cosθ.Solution .........Example 2.24 Find the entire area within the cardioid of Example 2.23.Solution .........Example 2.25 Find the area enclosed by the rose curve r = cos2θ.Solution .........Example 2.26 Find the area of the region that is inside of the cardioid r = 4−4cosθ andoutside of the circle r = 6.Solution .........
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Chapter 1 Topics in Analytic Geometry

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