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\begin{document}

\begin{equation}
\label{eq:calka:potrojna}
\begin{split}
\iiint\limits_V x dv & = 
\iint\limits_D\left(x \int\limits_{0}^{2-2x-2y}dz \right) d\sigma 
= \iint\limits_D x \left(2-2x-2y \right) d\sigma =\\
\intertext{określamy granice całkowania i obliczamy całkę względem $y$}
& =  \iint\limits_D \left(2x-2x^2-2xy \right) d\sigma = 
\int\limits_0^1 \left(\int\limits_0^{1-x}(2x-2x^2-2xy)dy \right) dx =\\
& = \int\limits_0^1 \biggl[2xy-2x^2y-xy^2 \biggr]_{y=0}^{y=1-x} dx =\\
& = \int\limits_0^1 \left(2x(1-x)-2x^2(1-x)-x(1-x)^2\right) dx =\\
\intertext{w końcu obliczamy całkę względem $x$}
& = \int\limits_0^1(x^3 -2x^2 +x) dx = 
\left[\frac{1}{4}x^4 -\frac{2}{3}x^3 +\frac{1}{2}x^2 \right]_0^1 =\\
& = \frac{1}{4} -\frac{2}{3} +\frac{1}{2} = \frac{1}{12}
\end{split}
\end{equation}

\end{document}