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If $f(x)$, $g(y)$, and $h(z)$ are real-valued functions of a single variable, does the following always hold? Is this the case for numerical approximations of the integral using quadrature?

$$ w(x,y,z) = \int\limits_{{x_1}}^{{x_2}} {\int\limits_{{y_1}}^{{y_2}} {\int\limits_{{z_1}}^{{z_2}} {f(x)g(y)h{{(z)}^{}}\,d{x^{}}\,d{y^{}}\,dz} } }$$

$$w(x,y,z) = \left[ {\int\limits_{{x_1}}^{{x_2}} {f(x)\,dx} } \right]\;\left[ {\int\limits_{{y_1}}^{{y_2}} {g(y)\,dy} } \right]\;\left[ {\int\limits_{{z_1}}^{{z_2}} {h(z)\,dz} } \right]$$

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    $\begingroup$ "Yes" for the the first question, assuming all the functions are integrable. $\endgroup$ – David Mitra Jan 19 '14 at 23:17
  • $\begingroup$ Thanks - this makes sense; it is also nice to have the edit for the formatting (good looking integrals). $\endgroup$ – Nicholas Kinar Jan 19 '14 at 23:20

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