What is the area of loop of the curve r = a sin 3 θ ?

Step-by-step solution

If curve r = a sin 3 θ , to trace the curve, we consider the following table: 3 θ = 0 π 2 π 3 π 2 2 π 5 π 2 3 π θ = 0 π 6 π 3 π 2 2 π 3 5 π 6 π r = 0 a 0 - a 0 a 0 Thus, there is a loop between θ = 0 to θ = π 3 and r varies from r = 0 to r = a . Hence, the area of the loop lying in the positive quadrant = 1 2 ∫ 0 π 3 r 2 d θ = a 2 2 ∫ 0 π 3 sin 2 3 θ d θ On putting, 3 θ = ϕ &#8658