Find the mass and center of mass of the lamina that occupies the region $D$ and has the given density function $\rho$.
$$D = \{(x, y) | 0 \leq x \leq 1, −1 \leq y \leq 1\}; \rho(x, y) = 7xy^2$$
I got my mass to be $7/3$ but I don't know how to go about finding the center of mass
Best Answer
By symmetry the $y$-component of the centre of mass is $0$. For the $x$-component, we find the moment of the lamina about the $y$-axis, and divide by the mass.
The moment about the $y$-axis is equal to $$\iint_D (x)(7xy^2)\,dy\,dx,$$ where $D$ is the rectangle $0\le x\le 1$, $-1\le y\le -1$. This integral can be evaluated using the same technique as the one you used to compute the mass.