Volume of a Region Defined by Inequalities in 3D

The question specifies a three-dimensional region via algebraic inequalities and asks for its volume without involving revolution or standard geometric solid formulas.

gaokao 2019 Q8 View
8. Two unit vectors $e _ { 1 } , e _ { 2 }$ have an angle of $60 ^ { \circ }$ between them. Vector $m = t e _ { 1 } + 2 e _ { 2 } ( t < 0 )$. Then
A. The maximum value of $\frac { | m | } { t }$ is $\frac { \sqrt { 3 } } { 2 }$
B. The minimum value of $\frac { | m | } { t }$ is $- 2$
C. The minimum value of $\frac { | m | } { t }$ is $\frac { \sqrt { 3 } } { 2 }$
D. The maximum value of $\frac { | m | } { t }$ is $- 2$
isi-entrance 2021 Q7 View
The volume of the region $S = \{ ( x , y , z ) : | x | + 2 | y | + 3 | z | \leq 6 \}$ is
(A) 36 .
(B) 48 .
(C) 72 .
(D) 6 .
todai-math 2023 Q4 View
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 $V$.