I don't even know where to start :(
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The area is given by ∫ (1/2) r^2 dθ.
In this case, we obtain
∫(θ = 0 to π) (1/2) (7 + sin(5θ))^2 dθ
= (1/2) ∫(θ = 0 to π) (49 + 14 sin(5θ) + sin^2(5θ)) dθ
= (1/2) ∫(θ = 0 to π) [49 + 14 sin(5θ) + (1/2)(1 - cos(10θ))] dθ
= (1/2) [49θ - (14/5) cos(5θ) + (1/2)(θ - sin(10θ)/10)] {for θ = 0 to π}
= (1/2) [(49π + 14/5 + π/2) + (14/5)]
= (1/20) (495π + 56).
I hope this helps!
In this case, we obtain
∫(θ = 0 to π) (1/2) (7 + sin(5θ))^2 dθ
= (1/2) ∫(θ = 0 to π) (49 + 14 sin(5θ) + sin^2(5θ)) dθ
= (1/2) ∫(θ = 0 to π) [49 + 14 sin(5θ) + (1/2)(1 - cos(10θ))] dθ
= (1/2) [49θ - (14/5) cos(5θ) + (1/2)(θ - sin(10θ)/10)] {for θ = 0 to π}
= (1/2) [(49π + 14/5 + π/2) + (14/5)]
= (1/20) (495π + 56).
I hope this helps!