1. **Problem Statement:** We need to find the area on stage within the optimal pickup range of the microphone, which is described by the cardioid given in polar coordinates as $$r = 8 + 8 \sin \theta$$ with the microphone at the pole. 2. **Formula for Area in Polar Coordinates:** The area enclosed by a polar curve $$r(\theta)$$ from $$\theta = a$$ to $$\theta = b$$ is given by: $$\text{Area} = \frac{1}{2} \int_a^b r(\theta)^2 \, d\theta$$ 3. **Important Notes:** - The cardioid is symmetric and completes one full loop as $$\theta$$ goes from $$0$$ to $$2\pi$$. - We will integrate over $$0$$ to $$2\pi$$ to find the total area. 4. **Set up the integral:** $$\text{Area} = \frac{1}{2} \int_0^{2\pi} (8 + 8 \sin \theta)^2 \, d\theta$$ 5. **Expand the square:** $$ (8 + 8 \sin \theta)^2 = 64 + 128 \sin \theta + 64 \sin^2 \theta $$ 6. **Rewrite the integral:** $$\text{Area} = \frac{1}{2} \int_0^{2\pi} \left(64 + 128 \sin \theta + 64 \sin^2 \theta\right) d\theta$$ 7. **Split the integral:** $$\text{Area} = \frac{1}{2} \left[ \int_0^{2\pi} 64 \, d\theta + \int_0^{2\pi} 128 \sin \theta \, d\theta + \int_0^{2\pi} 64 \sin^2 \theta \, d\theta \right]$$ 8. **Evaluate each integral:** - $$\int_0^{2\pi} 64 \, d\theta = 64 \times 2\pi = 128\pi$$ - $$\int_0^{2\pi} 128 \sin \theta \, d\theta = 0$$ (since $$\sin \theta$$ is periodic and symmetric over $$0$$ to $$2\pi$$) - For $$\int_0^{2\pi} 64 \sin^2 \theta \, d\theta$$, use the identity $$\sin^2 \theta = \frac{1 - \cos 2\theta}{2}$$: $$\int_0^{2\pi} 64 \sin^2 \theta \, d\theta = 64 \int_0^{2\pi} \frac{1 - \cos 2\theta}{2} d\theta = 32 \int_0^{2\pi} (1 - \cos 2\theta) d\theta$$ 9. **Evaluate the last integral:** $$32 \left[ \int_0^{2\pi} 1 \, d\theta - \int_0^{2\pi} \cos 2\theta \, d\theta \right] = 32 \left[ 2\pi - 0 \right] = 64\pi$$ 10. **Sum all parts:** $$\text{Area} = \frac{1}{2} \left(128\pi + 0 + 64\pi\right) = \frac{1}{2} \times 192\pi = 96\pi$$ 11. **Final answer:** The area within the optimal pickup range of the microphone is $$\boxed{96\pi}$$ square meters. This result uses the formula for area in polar coordinates and trigonometric identities to simplify the integral.