Skip to content

Commit

Permalink
京都大学 情報学研究科 知能情報学専攻 2023年8月実施 情報学基礎 F1-2 fix
Browse files Browse the repository at this point in the history
  • Loading branch information
Myyura committed Dec 25, 2024
1 parent aa72b9b commit 9ad596f
Showing 1 changed file with 36 additions and 3 deletions.
39 changes: 36 additions & 3 deletions docs/kakomonn/kyoto_university/informatics/ist_202308_kiso_f1_2.md
Original file line number Diff line number Diff line change
Expand Up @@ -7,7 +7,7 @@ tags:
# 京都大学 情報学研究科 知能情報学専攻 2023年8月実施 情報学基礎 F1-2

## **Author**
[Isidore](https://github.com/heacsing), Casablanca
[Isidore](https://github.com/heacsing), Casablanca, 祭音Myyura

## **Description**
### 設問1
Expand Down Expand Up @@ -38,10 +38,43 @@ $3x^2 + 2y^2 + z^2 = 1$ の条件の下で、$xyz$ の最大値と最小値を
## **Kai**
### 設問1
#### (1)
Perform the substitution $x = \tan \theta$, the answer is $\pi$
Let $x = \tan \theta$, we have $dx=\frac{d\theta}{\cos^{2} \theta}$. Then

$$
\begin{aligned}
\int_{0}^{\infty}\frac{1}{(1+x^{2})^{2}}dx
&= \int_{0}^{\pi/2}\cos^{4} \theta~\frac{d\theta}{\cos^{2} \theta}
= \int_{0}^{\pi/2}\cos^{2}\theta~d\theta \\
&= \frac{1}{2} \int_{0}^{\pi/2} (1 + \cos 2\theta) d\theta = \frac{\pi}{4}
\end{aligned}
$$

#### (2)
Perform the substitution $x = r \cos \theta, y = 2r \sin \theta$, the Jacobian determinant is $2r$ and the answer is $\frac{1}{3}\pi$
Let $x = r \cos \theta, y = 2r \sin \theta$, the Jacobian determinant

$$
\begin{aligned}
J &=
\begin{vmatrix}
\cos\theta & -r\sin\theta\\
2\sin\theta & 2r\cos\theta
\end{vmatrix}
= 2r\cos^{2}\theta+2r\sin^{2}\theta = 2r
\end{aligned}
$$

Then we have

$$
\begin{aligned}
\iint_{D}x^{2}y^{2}dxdy
&= \int_{0}^{1}\int_{0}^{2\pi}4r^{4}\sin^{2}\theta\cos^{2}\theta~(|2r|dr)d\theta
= \int_{0}^{1}2r^{5}~dr\int_{0}^{2\pi}\sin^{2}2\theta~d\theta\\[0.7em]
&= \int_{0}^{1}2r^{5}~dr\frac{1}{2}\int_{0}^{2\pi}(1-\cos 4\theta)~d\theta
= \left[\frac{r^{6}}{3}\right]_{0}^{1}\cdot \frac{1}{2}\left[\theta+\frac{1}{4}\sin 4\theta\right]_{0}^{2\pi}
= \frac{\pi}{3}
\end{aligned}
$$

### 設問2
#### (1)
Expand Down

0 comments on commit 9ad596f

Please sign in to comment.