Properties of compactly supported continuous function Let $a,b,c \in R$
Let D = {$(x_1,x_2,x_3): x_1^2 + x_2^2 +x_3^2 \leq 1  $}.
Let E = {$(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  $} and 
$\mathbf{A}=\left[\begin{array}{*{20}{c}}
\mathbf{a}&{0}&{0}\\
{0}&\mathbf{b}&{0}\\
{0}&{0}&\mathbf{c}
\end{array}\right]$ , det A $>1$.
Then for a compactly supported continuous function f on $R^3$, 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_{R^3} f(Ax)dx = \frac{1}{abc} \int_{R^3} f(x)dx $

Now, $\int_D f((ax_1,bx_2,cx_3))dx_1dx_2dx_3 = \frac{1}{abc}\int_D f((ax_1,bx_2,cx_3))adx_1 bdx_2 cdx_3  $
$ = \frac{1}{abc}\int_E f((x_1,x_2,x_3))dx_1dx_2dx_3 = \frac{1}{abc}\int_E f(x)dx $ 
Again,  $\int_{R^3} f((ax_1,bx_2,cx_3))dx_1dx_2dx_3 = \frac{1}{abc}\int_{R^3} f((ax_1,bx_2,cx_3))adx_1 bdx_2 cdx_3  $
$ = \frac{1}{abc}\int_{R^3} f((x_1,x_2,x_3))dx_1dx_2dx_3 = \frac{1}{abc}\int_{R^3} f(x)dx $ 
Hence, (3) and (4) are true.
