I'm attempting to evaluate the following triple integral:

$$ \int_{-1}^1 \int_{-1}^1 \int_0^{\frac{4}{\sqrt{2}}\big(1+yz-|y+z|\big)} \sqrt{2x^2+(y-z)^2} \, dx \, dy \, dz $$

I initially attempted a direct solution, and while the inner integral has an antiderivative, the upper limit poses challenges due to its complexity. Integrating the resulting expression with this limit inflates the integral. Subsequently, I explored a suitable transformation but haven't identified an effective one.

I'm seeking guidance on potential strategies or clever approaches to solve this integral more efficiently. Your insights and assistance would be greatly appreciated.

  • $\begingroup$ Split the integral over $y$ into two parts according to the sign of $y+z$. $\endgroup$
    – joriki
    Feb 7 at 19:20
  • $\begingroup$ Substituting $(x,y,z)=\left(u,\dfrac{v+w}{\sqrt2},\dfrac{-v+w}{\sqrt2}\right)$ (rotation transform) at the very least simplifies the integrand to $\sqrt2\sqrt{u^2+v^2}$. Unfortunately it makes a bigger mess of the limits and requires splitting up the region:$$4\sqrt2\left\{\int_0^\tfrac1{\sqrt2}\int_0^{\sqrt{w^2+4}-\sqrt2}+\int_\tfrac1{\sqrt2}^{\sqrt2}\int_0^{\sqrt2-w}\right\}\int_0^{4\sqrt2-\sqrt2(v+\sqrt2)^2+\sqrt2\,w^2}\sqrt{u^2+v^2}\,du\,dv\,dw$$ $\endgroup$
    – user170231
    Feb 8 at 1:21


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