Question

what we should do to find the area of a rose petal​

Answer 1

Answer:

Thus, the area of a petal is 1 2 ∫ − π / 6 π / 6 cos ⁡ 2 ( 3 θ ) d θ = 1 2 ( θ 2 + sin ⁡ ( 6 x ) 12 ) ∣ − π / 6 π / 6 = π 12 .

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Answer 2

A sketch is useful here, but the only important observation is that r=0 when θ=0, and again at π3. These are your limits for one petal.

Since the area of a polar curve between the rays θ=a and θ=b is given by ∫ba12r2dθ, we have

A=∫π/3012sin2(3θ)dθ=12∫π/301−cos(6θ)2dθ

=14[θ−sin(6θ)2]π/30=14(π3−12sin(6π3))=π12

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