There is a wooden block where $ACFD$ and $ABED$ are two congruent isosceles trapezoids, and $BCFE$ is a rectangle. Let the projection of point $A$ on line $BC$ be $M$ and its projection on plane $BCFE$ be $P$. Given that $\overline{AD} = 30$, $\overline{CF} = 40$, $\overline{AP} = 15$, and $\overline{BC} = 10$. Place plane $BCFE$ on a horizontal table, and call any plane parallel to $BCFE$ a horizontal plane. The intersection of a horizontal plane at distance $x$ from $A$ (where $0 < x < 15$) with the wooden block is a rectangle of area $20x + \frac{4}{9}x^2$. Divide the line segment $\overline{AP}$ into $n$ equal parts, and denote the division points along the direction of vector $\overrightarrow{AP}$ as $A = P_0, P_1, \ldots, P_{n-1}, P_n = P$. For each segment $\overline{P_{k-1}P_k}$, consider the rectangular prism formed by taking the rectangle formed by the intersection of the horizontal plane passing through $P_k$ with this wooden block as the base and $\overline{P_{k-1}P_k}$ as the height. Please use this slicing method to write down the Riemann sum estimating the volume of this wooden block (no need to simplify), express the volume of this wooden block as a definite integral, and find its value. (Non-multiple choice question, 6 points)

What is the limit
$$\lim _ { n \rightarrow \infty } \frac { 3 } { n ^ { 2 } } \left( \sqrt { 4 n ^ { 2 } + 9 \times 1 ^ { 2 } } + \sqrt { 4 n ^ { 2 } + 9 \times 2 ^ { 2 } } + \cdots + \sqrt { 4 n ^ { 2 } + 9 \times ( n - 1 ) ^ { 2 } } \right)$$
which of the following definite integrals can represent?
(1) $\int _ { 0 } ^ { 3 } \sqrt { 1 + x ^ { 2 } } d x$
(2) $\int _ { 0 } ^ { 3 } \sqrt { 1 + 9 x ^ { 2 } } d x$
(3) $\int _ { 0 } ^ { 3 } \sqrt { 4 + x ^ { 2 } } d x$
(4) $\int _ { 0 } ^ { 3 } \sqrt { 4 + 9 x ^ { 2 } } d x$
(5) $\int _ { 0 } ^ { 3 } \sqrt { 4 x ^ { 2 } + 9 } d x$