Math equation typeset problem

You could use a \substack to stack items underneath an operator:

\documentclass{article}
\usepackage{amsmath}% http://ctan.org/pkg/amsmath
\newcommand{\limitint}{\mathop{\int}}
\begin{document}
\begin{align}
  \limitint_{\substack{\text{text 1}\\\text{text 2}}} (\vec{\nabla} \times \vec{C})_n \,\mathrm{d}\vec{S} &= 
    \limitint_{\substack{\text{text 3}\\\text{text 4}}} \vec{\nabla} \cdot (\vec{\nabla} \times \vec{C}) \,\mathrm{d}V.
\end{align}
\end{document}

Depending on what you have written for text 1...text 4, you may want to use mathtools' \mathclap to avoid too much horizontal spacing:

\documentclass{article}
\usepackage{mathtools}% http://ctan.org/pkg/mathtools
\newcommand{\limitint}{\mathop{\int}}
\begin{document}
\begin{align}
  \limitint_{\mathclap{\substack{\text{text 1}\\\text{text 2}}}} (\vec{\nabla} \times \vec{C})_n \,\mathrm{d}\vec{S} &= 
    \limitint_{\mathclap{\substack{\text{text 3}\\\text{text 4}}}} \vec{\nabla} \cdot (\vec{\nabla} \times \vec{C}) \,\mathrm{d}V.
\end{align}
\end{document}

I introduce \stackint to stack any number of layers below the \displaystyle integral sign. No need to wrap the argument in \text macros. Each line of the undertext will be centered relative to the integral sign.

\documentclass{article}
\usepackage[usestackEOL]{stackengine}[2013-10-15]
\newcommand\stackint[1]{\tiny\Shortunderstack{{\normalsize$\displaystyle\int$} \\ #1}}
\begin{document}
\[
  \stackint{text 1\\text 2} (\vec{\nabla} \times \vec{C})_n \,\mathrm{d}\vec{S} = 
  \stackint{text 3\\text 4} \vec{\nabla} \cdot (\vec{\nabla} \times \vec{C}) \,\mathrm{d}V.
\]
\end{document}

To not let the under-text affect the integral width, apply this line in the preamble sometime after stackengine is loaded:

\def\useanchorwidth{T}