The graphs of the functions $f$ and $g$ are shown in the figure for $0 \leq x \leq 3$. It is known that $g(x) = \frac{12}{3 + x}$ for $x \geq 0$. The twice-differentiable function $f$, which is not explicitly given, satisfies $f(3) = 2$ and $\int_{0}^{3} f(x)\, dx = 10$. (a) Find the area of the shaded region enclosed by the graphs of $f$ and $g$. (b) Evaluate the improper integral $\int_{0}^{\infty} (g(x))^{2}\, dx$, or show that the integral diverges. (c) Let $h$ be the function defined by $h(x) = x \cdot f'(x)$. Find the value of $\int_{0}^{3} h(x)\, dx$.
The graphs of the functions $f$ and $g$ are shown in the figure for $0 \leq x \leq 3$. It is known that $g(x) = \frac{12}{3 + x}$ for $x \geq 0$. The twice-differentiable function $f$, which is not explicitly given, satisfies $f(3) = 2$ and $\int_{0}^{3} f(x)\, dx = 10$.
(a) Find the area of the shaded region enclosed by the graphs of $f$ and $g$.
(b) Evaluate the improper integral $\int_{0}^{\infty} (g(x))^{2}\, dx$, or show that the integral diverges.
(c) Let $h$ be the function defined by $h(x) = x \cdot f'(x)$. Find the value of $\int_{0}^{3} h(x)\, dx$.