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Let $a,b,c$ be positive real numbers, $$D:=\{(x_1,x_2,x_3)\in R^3:x_1^2+x_2^2+x_3^2\leq1\}, \\E:=\{(x_1,x_2,x_3)\in R^3:\frac{x_1^2}{a^2}+ \frac{x_2^2}{b^2}+\frac{x_3^2}{c^2}\leq1\}$$ and $$A=diag\left [{\begin{array}{cc} a & b&c \ \ \end{array} } \right]$$ with det$A>1$. Then, for a compactly supported continuous function $f$ on $R^3$, which of the following is correct?$$1.\int_D f(Ax)dx=\int_Ef(x)dx \\2.\int_D f(Ax)dx=\frac{1}{abc}\int_Df(x)dx\\3.\int_D f(Ax)dx=\frac{1}{abc}\int_Ef(x)dx\\4.\int_{R^3} f(Ax)dx=\frac{1}{abc}\int_{R^3}f(x)dx$$ My approach here is to find some easy example of compactly supported function on $R^3$ and try to see each option for its validity but I am not able to find any simple one except those bump types. Other idea is to look for some nice theorem on integrals that could help here but there too I don't find any. (I am not good in latex typing, so for any mistakes or not be able to follow the standard typing patterns I apologize.) (Thanks for any help )

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  • $\begingroup$ Your typesetting is fine. $\endgroup$ – zhw. Oct 31 '16 at 18:35
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The nonsingular linear transformation $A$ maps $D$ onto $E.$ Thus

$$\int_E f = \int_D f\circ A |\det A|$$

for any $f$ continuous on $E.$ This follows from the standard change of variables formula in several variables. This shows $3$ is true. Therefore $1,2$ don't hold. I'll let you think about 4.

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