As shown in the figure, consider a rectangular stone block with a vertex $A$ and a face containing point $A$. Let the midpoints of the edges of this face be $B , E , F , D$ respectively. Another face of the rectangular block containing point $B$ has its edge midpoints as $B , C , H , G$ respectively. Given that $\overline { B C } = 8$ and $\overline { B D } = \overline { D C } = 9$. The stone block is now cut to remove eight corners, such that the cutting plane for each corner passes through the midpoints of the three adjacent edges of that corner.
Find the length of $\overline { A D }$ and the volume of tetrahedron $A B C D$, and find the height from vertex $A$ to the base plane when $\triangle B C D$ is the base of the tetrahedron. (Volume of pyramid $= \frac { \text{Base area} \times \text{Height} } { 3 }$) (Non-multiple choice question, 8 points)

Let $f(x) = 3ax^{2} + (1 - a)$ be a real coefficient polynomial function, where $-\frac{1}{2} \leq a \leq 1$. On the coordinate plane, let $\Gamma$ be the region enclosed by $y = f(x)$ and the $x$-axis for $-1 \leq x \leq 1$.
Let $V$ be the volume of the solid of revolution obtained by rotating $\Gamma$ about the $x$-axis. For all $a \in \left[-\frac{1}{2}, 1\right]$, is $V$ always equal? If equal, find its value; if not equal, find the value of $a$ for which $V$ has a maximum value, and find this maximum value. (Non-multiple choice question, 6 points)

Consider a twice differentiable function $y ( x )$ in an $x y$ plane which connects two points $A ( - 1,2 )$ and $B ( 1,2 )$. Let $S$ be outer surface area of the cylindrical object created by rotation of the curve $y ( x )$ about the $x$ axis. Answer the following questions.
(1) Prove that the surface area $S$ is given by
$$\begin{aligned} S & = 2 \pi \int _ { - 1 } ^ { 1 } F \left( y , y ^ { \prime } \right) \mathrm { d } x \\ F \left( y , y ^ { \prime } \right) & = y \sqrt { 1 + \left( y ^ { \prime } \right) ^ { 2 } } \end{aligned}$$
where $y ^ { \prime } = \frac { \mathrm { d } y } { \mathrm {~d} x }$.
(2) Let the curve $y ( x )$ satisfy the following Euler-Lagrange equation for arbitrary $x$:
$$\frac { \partial F } { \partial y } - \frac { \mathrm { d } } { \mathrm {~d} x } \frac { \partial F } { \partial y ^ { \prime } } = 0$$
Considering Eq. (2.3) along with $\frac { \mathrm { d} F } { \mathrm {~d} x }$, prove that the following relation holds:
$$F - y ^ { \prime } \frac { \partial F } { \partial y ^ { \prime } } = c$$
Here $c$ is a constant.
(3) Express a differential equation satisfied by the curve $y ( x )$ using $y , y ^ { \prime } , c$.
(4) Represent the curve $y ( x )$ as a function of $x$ and $c$.
Obtain an equation which should be satisfied by the constant $c$.

In the three-dimensional orthogonal $x y z$ coordinate system, consider the region $V$ that satisfies Equations (1) and (2).
$$\begin{aligned} & x ^ { 2 } + y ^ { 2 } - z ^ { 2 } \geq 0 \\ & x ^ { 2 } + y ^ { 2 } + 2 x \leq 0 \end{aligned}$$
I. Sketch the cross-sectional shape of the region $V$ at $z = 1$.
II. Obtain the surface area of the region $V$.

In the analytic plane; the region bounded by the x-axis, the line $x + y = 2$, and the curve $y = \sqrt { x }$ is rotated $360 ^ { \circ }$ around the x-axis.
What is the volume of the solid of revolution obtained in cubic units?
A) $\frac { \pi } { 2 }$
B) $\frac { 2 \pi } { 3 }$
C) $\frac { 3 \pi } { 4 }$
D) $\frac { 5 \pi } { 6 }$
E) $\frac { 7 \pi } { 6 }$

In the first quadrant; the region between the x-axis, the hyperbola $x ^ { 2 } - y ^ { 2 } = 1$, and the circle $x ^ { 2 } + y ^ { 2 } = 7$ is rotated $360 ^ { \circ }$ around the x-axis.
Which of the following is the integral expression of the volume of the solid of revolution obtained?
A) $\pi \int _ { 1 } ^ { 2 } \left( x ^ { 2 } - 1 \right) d x + \pi \int _ { 2 } ^ { \sqrt { 7 } } \left( 7 - x ^ { 2 } \right) d x$
B) $\pi \int _ { 1 } ^ { 2 } \left( x ^ { 2 } + 1 \right) d x + \pi \int _ { 2 } ^ { \sqrt { 7 } } \left( 7 + x ^ { 2 } \right) d x$
C) $\pi \int _ { 1 } ^ { 2 } \left( x ^ { 2 } - 1 \right) d x + \pi \int _ { 2 } ^ { \sqrt { 7 } } ( 7 - x ) ^ { 2 } d x$
D) $\pi \int _ { 1 } ^ { \sqrt { 3 } } ( x - 1 ) ^ { 2 } d x + \pi \int _ { \sqrt { 3 } } ^ { \sqrt { 7 } } \left( 7 + x ^ { 2 } \right) d x$
E) $\pi \int _ { 1 } ^ { \sqrt { 3 } } ( x - 1 ) ^ { 2 } d x + \pi \int _ { \sqrt { 3 } } ^ { \sqrt { 7 } } ( 7 - x ) ^ { 2 } d x$

In the rectangular coordinate plane, the region between the parabola $y = x ^ { 2 }$, the line $x = 1$, and the line $y = 0$ is shown.
What is the volume in cubic units of the solid of revolution obtained by rotating this region $360 ^ { \circ }$ about the line $\mathbf { y = - 1 }$?
A) $\frac { 3 \pi } { 4 }$
B) $\frac { 5 \pi } { 8 }$
C) $\frac { 7 \pi } { 10 }$
D) $\frac { 11 \pi } { 12 }$
E) $\frac { 13 \pi } { 15 }$

The position of two iron balls in the shape of spheres with radius 3 units placed inside a right circular cylinder with radius 6 units is shown in Figure 1.
The cylinder is filled with water until both balls are completely submerged in water and the view in Figure 2 is obtained.
Accordingly, what is the volume of water in the cylinder in Figure 2 in cubic units?
A) $96 \pi$
B) $108 \pi$
C) $120 \pi$
D) $132 \pi$
E) $144 \pi$

In the rectangular coordinate plane, the region between the lines $y = - x + 5$, $y = x + 3$ and the coordinate axes is shown below.
What is the volume of the solid of revolution obtained by rotating this region $360 ^ { \circ }$ about the y-axis?
A) $37 \pi$
B) $38 \pi$
C) $40 \pi$
D) $41 \pi$
E) $42 \pi$

A square right prism with edge lengths 10, 10, 25 units is divided into unit cubes. Then, using all of these cubes, a square right prism with height 1 unit is formed with no gaps between them.
Accordingly, what is the surface area of this square right prism in square units?
A) 5200 B) 5400 C) 5600 D) 5800 E) 6000

A right circular cone with height 10 units is placed inside a hollow right circular cylinder with height 10 units as shown in Figure 1. Water with volume $\mathrm { V } _ { 1 }$ cubic units is poured between this cylinder and cone, and the water height becomes 5 units. Then this object is inverted as shown in Figure 2, and after adding more water, the water volume becomes $\mathrm { V } _ { 2 }$ cubic units and the height becomes 5 units.
Accordingly, what is the ratio $\frac { \mathrm { V } _ { 1 } } { \mathrm {~V} _ { 2 } }$?
(During this process, water does not enter the cone.)
A) $\frac { 3 } { 7 }$ B) $\frac { 5 } { 11 }$ C) $\frac { 8 } { 15 }$ D) $\frac { 10 } { 21 }$ E) $\frac { 15 } { 31 }$