Volume Elipsoida

[cotrans]

Shalom aleichem.

\section{Volume Elipsoida}

Ada sebuah elipsoida pejal, yaitu
\[ E := \{(x, y, z) \in \mathbb{R}^3 ~|~ (x/a)^2 + (y/b)^2 + (z/c)^2 \leq 1 \} \]
di mana $a, b, c \in \mathbb{R}^+$ adalah setengah dari panjang sumbu-sumbu utama elipsoida tersebut.

Pertidaksamaan elipsoida tersebut dapat ditulis sebagai sebuah persamaan, yaitu
\[ (x/a)^2 + (y/b)^2 + (z/c)^2 = r^2 \]
di mana $r \in [0, 1]$.

Salah satu parameterisasi dari $E$ tersebut adalah
\[ x = ar\sin\alpha\cos\beta, \]
\[ y = br\sin\alpha\sin\beta, \]
\[ z = cr\cos\alpha. \]
Volume dari $E$ adalah
\[ V := \int_E |dx\wedge dy\wedge dz| \]
di mana
\[ dx\wedge dy\wedge dz = \begin{vmatrix} \partial x/\partial r & \partial x/\partial\alpha & \partial x/\partial\beta \\ \partial y/\partial r & \partial y/\partial\alpha & \partial y/\partial\beta \\ \partial z/\partial r & \partial z/\partial\alpha & \partial z/\partial\beta \end{vmatrix}dr\wedge d\alpha\wedge d\beta \]
\[ = \begin{vmatrix} a\sin\alpha\cos\beta & ar\cos\alpha\cos\beta & -ar\sin\alpha\sin\beta \\ b\sin\alpha\sin\beta & br\cos\alpha\sin\beta & br\sin\alpha\cos\beta \\ c\cos\alpha & -cr\sin\alpha & 0 \end{vmatrix}dr\wedge d\alpha\wedge d\beta \]
\[ = abcr^2\begin{vmatrix} \sin\alpha\cos\beta & \cos\alpha\cos\beta & -\sin\alpha\sin\beta \\ \sin\alpha\sin\beta & \cos\alpha\sin\beta & \sin\alpha\cos\beta \\ \cos\alpha & -\sin\alpha & 0 \end{vmatrix}dr\wedge d\alpha\wedge d\beta \]
\[ = abcr^2(\cos^2\alpha\sin\alpha\cos^2\beta + \sin^3\alpha\sin^2\beta + \sin^3\alpha\cos^2\beta \]
\[ + \sin\alpha\cos^2\alpha\sin^2\beta)dr\wedge d\alpha\wedge d\beta \]
\[ = abcr^2(\cos^2\alpha\sin\alpha + \sin^3\alpha)dr\wedge d\alpha\wedge d\beta \]
\[ = abcr^2(\cos^2\alpha + \sin^2\alpha)\sin\alpha\,dr\wedge d\alpha\wedge d\beta \]
\[ = abcr^2\sin\alpha\,dr\wedge d\alpha\wedge d\beta \]
sehingga
\[ V = abc\int_0^{2\pi} \int_0^\pi \int_0^1 r^2\sin\alpha\,dr\,d\alpha\,d\beta = \frac{4}{3}\pi abc. \]

Nderek langkung.