Mathematics

The key is recognizing that a petal of $r = a\sin(n\theta)$ traces out when $r$ goes from 0 to its maximum and back to 0 as $\theta$ sweeps through one complete cycle. Use the polar area formula $A = \frac{1}{2}\int r^2 \, d\theta$ over the angular interval for a single petal, then exploit the symmetry and periodicity of the sine function to simplify the integral into a standard form you can evaluate.

The key is recognizing that a petal of $r = a\sin(n\theta)$ traces out when $r$ goes from 0 to its maximum and back to 0 as $\theta$ sweeps through one complete cycle. Use the polar area formula $A = \frac{1}{2}\int r^2 \, d\theta$ over the angular interval for a single petal, then exploit the symmetry and periodicity of the sine function to simplify the integral into a standard form you can evaluate.