one bound integral

Question

so i have a formula for finding the center of mass of a body:

$\frac{1}{m}\int_V\vec r\,dm$

what does it mean when an integral has only one bound like this on the bottom?

Answer 1

Remark that we can write $$\vec{R}_{cm}=\frac{1}{M}\int_V\rho(\vec{r})\,\vec{r}\,dV,$$ where we integrate over the volume $V$. If for example our region $V=[x_0,x_1]\times[y_0,y_1]\times[z_0,z_1]$, we write

$$ \vec{R}_{cm}=\frac{1}{M}\int_{x_0}^{x^1}\int_{y_0}^{y_1}\int_{z_0}^{z_1}\rho(x,y,z)\begin{pmatrix}x\\y\\z\\ \end{pmatrix}dx\,dy\,dz$$

Or one can use some different coordinates to calculate this integral, dependant on the volume.