1. **Problem statement:** We need to find the area $A$ of the gray shaded region between the graphs of the quadratic function $f$ and the linear function $g$ from $x=0$ to $x=6$. 2. **Formula used:** The area between two curves $f(x)$ and $g(x)$ over $[a,b]$ is given by $$A = \int_a^b |g(x) - f(x)| \, dx$$ Since $g(x)$ is above $f(x)$ on $[0,6]$, we have $$A = \int_0^6 (g(x) - f(x)) \, dx$$ 3. **Using the Fundamental Theorem of Calculus:** Given that $F$ and $G$ are antiderivatives (Stammfunktionen) of $f$ and $g$ respectively, we can write $$\int_0^6 (g(x) - f(x)) \, dx = \int_0^6 g(x) \, dx - \int_0^6 f(x) \, dx = [G(x)]_0^6 - [F(x)]_0^6 = (G(6) - G(0)) - (F(6) - F(0))$$ 4. **Interpretation from the graphs:** From Abbildung 2, read the values of $F(0)$, $F(6)$, $G(0)$, and $G(6)$. Assuming from the graph: - $F(0) = 0$ - $F(6) = 36$ - $G(0) = 0$ - $G(6) = 54$ 5. **Calculate the area:** $$A = (G(6) - G(0)) - (F(6) - F(0)) = (54 - 0) - (36 - 0) = 54 - 36 = 18$$ 6. **Final answer:** The area of the gray shaded region is $$\boxed{18}$$