Integral on sphere and ellipsoid Let $a,b,c \in \mathbb{R},$ $\mathbf{A}=\left[\begin{array}{*{20}{c}}
\mathbf{a}&{0}&{0}\\
{0}&\mathbf{b}&{0}\\
{0}&{0}&\mathbf{c}
\end{array}\right] , ~~\det A >1$
Let $~D = \{(x_1,x_2,x_3): x_1^2 + x_2^2 +x_3^2 \leq 1  \}~$ and
$~E = \left\{(x_1,x_2,x_3): \frac{x_1^2}{a^2} + \frac{x_2^2}{b^2} + \frac{x_3^2}{c^2} \leq 1  \right\}~.$
Then for a compactly supported continuous function $f$ on $\mathbb{R}^3$, could anyone tell me which of the following are correct?
1.$\int_D f(Ax)dx = \int_E f(x)dx  $
2.$\int_D f(Ax)dx = \frac{1}{abc} \int_D f(x)dx $
3.$\int_D f(Ax)dx = \frac{1}{abc} \int_E f(x)dx $
4.$\int_{\mathbb{R}^3} f(Ax)dx = \frac{1}{abc} \int_{\mathbb{R}^3} f(x)dx $
 A: A function $~f~$ is said to be compactly supported if it is zero outside a compact set.
Let $~x=(x_1,x_2,x_3)\in\mathbb R^3~,$ be any arbitrary vector.
$$\therefore~~Ax=\left[\begin{array}{*{20}{c}}
\mathbf{a}&{0}&{0}\\
{0}&\mathbf{b}&{0}\\
{0}&{0}&\mathbf{c}
\end{array}\right]\left[\begin{array}{*{20}{c}}
{x_1}\\{x_2}\\{x_3}
\end{array}\right]=(ax_1,bx_2,cx_3)$$
Now $$\int_D f(Ax)dx =\iiint_{x_1^2+x_2^2+x_3^2\le1} f(ax_1,bx_2,cx_3)~dx_1~dx_2~dx_3\tag1$$
Putting $~~ax_1=y_1,~~bx_2=y_2,~~cx_3=y_3   \implies x_1=\dfrac{y_1}{a},~~x_2=\dfrac{y_2}{b},~~x_3=\dfrac{y_3}{c}$ $$\implies dx_1=\dfrac{dy_1}{a},~~dx_2=\dfrac{dy_2}{b},~~dx_3=\dfrac{dy_3}{c}$$
So $$x_1^2+x_2^2+x_3^2\le1\implies \dfrac{y_1^2}{a^2}+\dfrac{y_2^2}{b^2}+\dfrac{y_3^2}{c^2}\le1$$
So from $(1)$, $$\int_D f(Ax)dx =\iiint_{\frac{y_1^2}{a^2}+\frac{y_2^2}{b^2}+\frac{y_3^2}{c^2}\le1} f(y_1,y_2,y_3)~\dfrac{dy_1}{a}~\dfrac{dy_2}{b}~\dfrac{dy_3}{c}$$
$$=\dfrac{1}{abc}\int_E f(y) dy=\dfrac{1}{abc}\int_E f(x) dx$$ $($ as $\det A>1$, therefore $abc>1$ $)$ 
Thus option $(3)$ is correct but option $(1)$ and $(2)$ are not correct.
Similarly, we can show that $$\int_{\mathbb R^3} f(Ax)dx =\dfrac{1}{abc}\int_{\mathbb R^3} f(x) dx$$ Hence option $(3)$ and option $(4)$ are the only correct options.
