todai-math 2023 Q4

todai-math · Japan · todai-engineering-math Volumes of Revolution Volume of a Region Defined by Inequalities in 3D

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 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 $V$.

Paper Questions

Q1 Q2 Q3 Q4 Q5 Q6