c10s5 - 10.5 Area and Lengths in Polar Coordinates The area...

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Unformatted text preview: 10.5 Area and Lengths in Polar Coordinates The area of a circular sector is A= 12 r θ. 2 The area of a polar region is A= 1 2 ∫a [ f (θ )] dθ b 2 or A=∫ b 1 a2 r 2 dθ with r = f (θ ) It can be helpful to graph the functions and to make use of the symmetry in many polar graphs. Example: Find the area of the region bounded by r = cos 3θ −π π ≤θ ≤ . 12 12 Example: Find the area of the region bounded by the curve r = 4 − 4 cos θ Example: Find the area of the region bounded by the curve r 2 = sin 2θ Example: Find the area of the region bounded by one loop of the curve . . r = 3 sin 2θ . Area Between Curves A= 1 2 ∫ ([ f (θ )] − [ g (θ )] )dθ b 2 2 a Example: Find the area of the region inside the first curve and outside the second curve. r = 3 cos θ , r = 2 − cos θ . Example: Find the area of the region inside both curves. r = 4 − 4 cos θ r = sin 2θ , r = sin θ . ...
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