1. **State the problem:** Evaluate the definite integral $$\int_{0}^{3 \pi} 90^2 \sin \left( \frac{1}{6} \theta \right) d\theta.$$\n\n2. **Recall the formula:** The integral of $$\sin(ax)$$ with respect to $$x$$ is $$-\frac{1}{a} \cos(ax) + C$$.\n\n3. **Identify constants:** Here, $$a = \frac{1}{6}$$ and the constant multiplier outside the sine function is $$90^2 = 8100$$.\n\n4. **Set up the integral:**\n$$\int 8100 \sin \left( \frac{1}{6} \theta \right) d\theta = 8100 \int \sin \left( \frac{1}{6} \theta \right) d\theta.$$\n\n5. **Integrate:**\n$$8100 \times \left(-6 \cos \left( \frac{1}{6} \theta \right) \right) + C = -48600 \cos \left( \frac{1}{6} \theta \right) + C.$$\n\n6. **Evaluate the definite integral:**\n$$\int_{0}^{3 \pi} 8100 \sin \left( \frac{1}{6} \theta \right) d\theta = \left[-48600 \cos \left( \frac{1}{6} \theta \right) \right]_{0}^{3 \pi}.$$\n\n7. **Calculate the cosine values:**\n$$\cos \left( \frac{1}{6} \times 3 \pi \right) = \cos \left( \frac{3 \pi}{6} \right) = \cos \left( \frac{\pi}{2} \right) = 0,$$\n$$\cos(0) = 1.$$\n\n8. **Substitute and simplify:**\n$$-48600 (0) - \left(-48600 (1)\right) = 0 + 48600 = 48600.$$\n\n**Final answer:** $$\boxed{48600}.$$