How to flush left centered part in multline environment?

Question

I would like to use multline environment, but in that way that centered part be left aligned. With multline centered part is centered, and I would like if we can somehow to aligned it to the left (but to stay centered). And of course, if this is possible to do with align environment (but that first line is flush to left and last to the right), that is better for me, because than I can label arbitrary equations (because with multline I can label just last one equation). I hope I didn't confuse you too much.

Here you MWE

\documentclass{article}
\usepackage{amsmath,amssymb}
\usepackage{lipsum}

\begin{document}

\lipsum[1]

\begin{multline}
 A = \lim _{n\rightarrow \infty }\Delta x\left( a^{2}+\left( a^{2}+2a\Delta x
 +\left( \Delta x\right) ^{2}\right)\right.\\
 +\left( a^{2}+2\cdot 2a\Delta x+2^{2}\left( \Delta x\right) ^{2}\right)\\
 +\left( a^{2}+2\cdot 3a\Delta x+3^{2}\left( \Delta x\right) ^{2}\right)\\
 + \ldots\\
 \left.+\left( a^{2}+2\cdot (n-1)a\Delta x +(n-1)^{2}\left( \Delta x\right) ^{2}\right) \right)\\
 = \frac{1}{3}\left( b^{3}-a^{3}\right) + \text{something}
\end{multline}

\lipsum[2]

\end{document}

Answer 1

Perhaps you mean like this? (I used an subsidiary aligned for the inner alignment)

The code:

\documentclass{article}
\usepackage{amsmath,amssymb}
\usepackage{lipsum}

\begin{document}

\lipsum[1]

\begin{multline}
 A = \lim _{n\rightarrow \infty }\Delta x\left( a^{2}+\left( a^{2}+2a\Delta x
 +\left( \Delta x\right) ^{2}\right)\right.\\
\begin{aligned} 
 &+\left( a^{2}+2\cdot 2a\Delta x+2^{2}\left( \Delta x\right) ^{2}\right)\\
 &+\left( a^{2}+2\cdot 3a\Delta x+3^{2}\left( \Delta x\right) ^{2}\right)\\
 &+ \cdots\\
 &+\left.\left( a^{2}+2\cdot (n-1)a\Delta x +(n-1)^{2}\left( \Delta x\right) ^{2}\right) \right)\\
\end{aligned} \\
 = \frac{1}{3}\left( b^{3}-a^{3}\right) + \text{something}
\end{multline}

\lipsum[2]

\end{document}

Instead of \left, \right, use the \big, \bigg, family of commands as in

\documentclass{article}
\usepackage{amsmath,amssymb}
\usepackage{lipsum}

\begin{document}

\lipsum[1]

\begin{multline}
 A = \lim _{n\rightarrow \infty }\Delta x \Bigl( a^{2}+\bigl( a^{2}+2a\Delta x
 + ( \Delta x ) ^{2}\bigr) \\
\begin{aligned} 
 &+ \bigl( a^{2}+2\cdot 2a\Delta x+2^{2} ( \Delta x ) ^{2} \bigr)\\
 &+ \bigl( a^{2}+2\cdot 3a\Delta x+3^{2} ( \Delta x ) ^{2} \bigr)\\
 &+ \cdots\\
 &+\bigl( a^{2}+2\cdot (n-1)a\Delta x +(n-1)^{2} ( \Delta x ) ^{2} \bigr) \Bigr)\\
\end{aligned} \\
 = \frac{1}{3} ( b^{3}-a^{3} ) + \text{something}
\end{multline}

\lipsum[2]

\end{document}

Even most of the parentheses can be safely removed.