Expected value of a bivariate distribution as an integral Let's assume an absolutely continuous random variable, $X$, with PDF $f(x)$.
$$\mathbb{E}\big[X\big] = \int_{\mathbb{R}}xf(x)dx$$
If $X\sim f(x_1,x_2)$ is multivariate, then it makes sense to me to use the following integral.
$$\int_{\mathbb{R}\times\mathbb{R}}x_1x_2f(x_1,x_2)d(x_1,x_2)$$
Then use Fubini's theorem and calculate the double integral to obtain a number.
However, I would consider the expected value to be $\big(\mathbb{E}\big[X_1], \mathbb{E}\big[X_2]\big)$, a vector, not a number.
Why should the two disagree?
 A: The straightforward extension of the univariate case
$$
\mathbb{E}\big[X\big] = \int_{\mathbb{R}}xf(x)dx
$$
to the bivariate one is
$$
\int_{\mathbb{R}\times\mathbb{R}}\color{blue}{(x_1,x_2)}f(x_1,x_2)d(x_1,x_2)
$$
rather than
$$
\int_{\mathbb{R}\times\mathbb{R}}\color{blue}{x_1x_2}f(x_1,x_2)d(x_1,x_2).
$$
While the notation might be unusual, it can be considered a shorthand for two integrals
$$
\left(\quad \int_{\mathbb{R}\times\mathbb{R}}\color{blue}{x_1}f(x_1,x_2)d(x_1,x_2), \quad \int_{\mathbb{R}\times\mathbb{R}}\color{blue}{x_2}f(x_1,x_2)d(x_1,x_2) \quad\right)
$$
that yield the desired $\big(\mathbb{E}\big[X_1], \mathbb{E}\big[X_2]\big)$, a vector.
A: If your random variable is bivariate, then every realization is a pair of numbers.
The expectation of a random number can be thought of as "the long-run average".
A long-run average of a large number of pairs makes most sense as a pair of numbers, not a single number. Specifically, as the pair of the separate long-run averages.
Which is why the expectation of a multivariate random variable is defined as the vector of separate expectations.

More mathematically, the definition is that
$$ E[X] = \int_\Omega X(\omega)\,dP(\omega), $$
and of course each $X(\omega)$ is a vector. There is no notion of multiplying the components of $X$ together.
This is just integrating a vector-valued function $f\colon X\to\mathbb{R}^k$ to have $\int_X f(x)\,dx$, which is a different animal than integrating the product of the components of $f$, which would be $\int_X\prod f_i(x)\,dx$.
A: If $X$ is a bivariate random variable $[X_1,X_2]$, then $X_1$ and $X_2$ are functions of $X$ (projections of $X$ onto the individual components). Thus, $E[X]$ is the vector $\big[E[X_1],E[X_2]\big]$ while what you are computing with the double integral that you wrote is $E[X_1X_2]$.
