Notation for integral of a vector function over an ellipsoid For a short proof, I need to write a point $\pmb y\in\mathbb{R}^p$ as 
 the integral of the surface of the ellipse $\pmb x^{\top}\pmb Q\pmb x=c$ where $\pmb Q$ is a $p$ by $p$ PSD matrix (for now $\pmb y$ is defined in words). What is the formal way to write this? Can we do better than:
$$\pmb y=\int_{\pmb x\in\mathbb{R}^p:\pmb x^{\top}\pmb Q\pmb x=c} \pmb x d(\pmb x)$$
Also, I am not a professional mathematician so I am not too sure about the $d(\pmb x)$ part. 

You have lots of options here, but a typical choice is to denote the ellipsoid itself by a symbol, say,
$$E := \{{\bf x} \in \Bbb R^p : {\bf x}^T {\bf Q} {\bf x} = c\},$$
and then write the integral the integral of a real-valued function $f$ as
$$\iint_E f \, dS.$$
Here, $dS$ is the infinitesimal area element of the surface $E$.
In our case, we're integrating a vector function (in fact, just the identity function ${\bf x} \mapsto {\bf x}$), and we can use essentially the same syntax and write
$${\bf y} = \iint_E {\bf x} \, dS.$$
(Of course, our $E$ is symmetric w.r.t. reflection through the origin, so in our case the integral is automatically zero.)
The double integral symbol simply reminds us that we're integrating over a surface (as to evaluate such an integral, often we parameterize the surface and then evaluate the appropriate double integral over the domain of the parameterization), but this emphasis is optional, as one will occasionally see $$\int_E f \, dS$$ instead. The ellipse is a closed surface, and some authors will remind you of by decorating the double integral $\iint$ with a loop that overlaps both of the integral symbols:
 .
(This symbol can produced with $\texttt{\oiint}$ with various $\LaTeX$ packages, but to my knowledge it is not supported by MathJax, which is used on this site for rendering.)
