A little stuck on what I assume is probably a fairly basic vector calculus integration problem, but I haven't been able to find any resources online that deal with multivariable integration in a way that jives with the notation I understand for this problem.
I'm trying to integrate a second order Taylor approximation in $\mathbb{R}^n$ so as to prove the formula for the Laplacian $$\Delta u(x) = \lim_{r \to 0} \frac{1}{(n+2) r^2 |B_r|} \int_{B_r(x)} u(y)-u(x) dy.$$
Here $B_r(x)$ is the $r$-ball around $x$, and $|B_r|$ is the volume of this ball.
The idea is to take the integrand on the right side, express it as a 2nd order taylor approximation around x, and then show that the RHS equals the Laplacian when evaluated. I can (somewhat informally) argue away the high order term as going to zero when integrated. But I get stuck trying to figure out what the integral of the other terms should be.
Currently I'm stuck trying to evaluate $\int_{B_r(0)} x_j dx$ and $\int_{B_r(0)} x_j x_k dx$, where $x_j, x_k$ are individual components of the vector $x$. Though maybe this is barking up the wrong tree and I need to figure out how to express the taylor series in such a way that the result of the integral has some cancellation with $r^2$ or $|B_r|$.
I'm pretty new to vector calculus, so any help would be appreciated.
EDIT: Based on suggestion, the first moment is zero always, and the second zero except when $j=k$. Since the term is actually $\frac{\partial^2 u(x)}{\partial x_j \partial x_k}\int_{B_r(0)} x_j x_k dx$, this means I just need to solve $\int_{B_r(0)} x_j^2 dx$ to find the coefficient on $\partial_jj u(x)$, which would hopefully turn out to be something that cancels somehow with $\lim_{r\to 0} \frac{1}{(n+2) r^2 |B_r|}$. Of course, being a vector calculus neophyte, I'm still not sure how to solve this integral.