Is it possible to do this subtraction ?
$\int\limits_{z = 0}^{z = \frac{b}{a}} {\int\limits_{x = 0}^{x = \infty } {\int\limits_{y = \frac{a}{x}}^{y = \frac{b}{{xz}}} {f\left( y \right)f\left( x \right)f\left( z \right)dydxdz = } } } \int\limits_{z = 0}^{z = \frac{b}{a}} {\int\limits_{x = 0}^{x = \infty } {\int\limits_{y = 0}^{y = \frac{b}{{xz}}} {f\left( y \right)f\left( x \right)f\left( z \right)dydxdz} } } - \int\limits_{z = 0}^{z = \frac{b}{a}} {\int\limits_{x = 0}^{x = \infty } {\int\limits_{y = 0}^{y = \frac{a}{x}} {f\left( y \right)f\left( x \right)f\left( z \right)dydxdz} } } $
where $a,b,f\left( y \right),f\left( x \right),f\left( z \right) \geqslant 0$ and ${f\left( y \right),f\left( x \right),f\left( z \right)}$ are all decaying exponential function.
If it wrong then why ? Thank you very much ?
I don't know what does "decaying exponential function" exactly mean, but one can claim that you are right if all the integrals converges. From the Linearity of integrals, we have: \begin{aligned} &\quad \int\limits_0^{\frac{b}{a}}{\int\limits_0^{\infty}{\int\limits_{\frac{a}{x}}^{\frac{b}{xz}}{f\left( y \right) f\left( x \right) f\left( z \right) dydxdz=}}}\int\limits_0^{\frac{b}{a}}{\int\limits_0^{\infty}{\left( \int\limits_{\frac{a}{x}}^{\frac{b}{xz}}{f\left( y \right) dy} \right) f\left( x \right) f\left( z \right) dxdz}} \\ &=\int\limits_0^{\frac{b}{a}}{\int\limits_0^{\infty}{\left( \int\limits_0^{\frac{b}{xz}}{f\left( y \right) dy-\int\limits_0^{\frac{a}{x}}{f\left( y \right) dy}} \right) f\left( x \right) f\left( z \right) dxdz}} \\ &=\int\limits_0^{\frac{b}{a}}{\int\limits_0^{\infty}{\left( \int\limits_0^{\frac{b}{xz}}{f\left( y \right) dy} \right) f\left( x \right) f\left( z \right) dxdz}}-\int\limits_0^{\frac{b}{a}}{\int\limits_0^{\infty}{\left( +\int\limits_0^{\frac{a}{x}}{f\left( y \right) dy} \right) f\left( x \right) f\left( z \right) dxdz}} \\ &=\int\limits_0^{\frac{b}{a}}{\int\limits_0^{\infty}{\int\limits_0^{\frac{b}{xz}}{f\left( y \right) f\left( x \right) f\left( z \right) dydxdz}}}-\int\limits_0^{\frac{b}{a}}{\int\limits_0^{\infty}{\int\limits_0^{\frac{a}{x}}{f\left( y \right) f\left( x \right) f\left( z \right) dydxdz}}} \end{aligned} As desired.