1. **State the problem:** Evaluate the definite integral $$\int_0^\pi \sin^2(t) \cos(t) \, dt$$. 2. **Use substitution:** Let $$u = \sin(t)$$, then $$du = \cos(t) dt$$. 3. **Change the limits:** When $$t=0$$, $$u=\sin(0)=0$$; when $$t=\pi$$, $$u=\sin(\pi)=0$$. 4. **Rewrite the integral:** The integral becomes $$\int_0^0 u^2 \, du$$. 5. **Evaluate the integral:** Since the limits are the same, the integral evaluates to $$0$$. **Final answer:** $$0$$.