The elemental area denoted by the shaded region is

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The elemental area, denoted by the shaded region, is found by taking the area to the the 3sin r θ = curve and subtract the area to the 2 sin r θ = curve. The area integral is thus: /2 2 2 /6 1 2 (3sin ) (2 sin ) 2 A d π π θ θ θ = Note that the factor “ 2 ” is used because we need to double the result to find the total area by virtue of symmetry. After some algebra, the integral becomes: /2 /2 /2 2 /6 /6 /6 8 sin 4 sin 4 A d d d π π π π π π θ θ θ θ θ = + You can complete the problem from here.
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