Mathematics

A rose curve traces out petals as the radius oscillates between positive and negative values. The key is to find where one petal starts and ends by identifying when $r = 0$, then integrate $\frac{1}{2}r^2$ over that angular interval. For $r = 2\sin 3\theta$, the petals are symmetric, so you can find one petal's bounds and use the polar area formula to capture its full shape.

A rose curve traces out petals as the radius oscillates between positive and negative values. The key is to find where one petal starts and ends by identifying when $r = 0$, then integrate $\frac{1}{2}r^2$ over that angular interval. For $r = 2\sin 3\theta$, the petals are symmetric, so you can find one petal's bounds and use the polar area formula to capture its full shape.