Question

The density of a thin plate is given by p(x, y) = xy and its mass by M. The plate is bounded by = √x, x = 1, and the x-axis. Set up the integral for the y-coordinate of the plate's center of y = mass.​

Answer 1

The $y$-coordinate of the center of mass is the average value of $y \rho(x,y)$ over the plate, given by the ratio of the integral of $y\rho(x,y)$ to the mass of the plate.

The mass of the plate is

$\displaystyle \int_0^1 \int_{y^2}^1 xy \, dx \, dy$

while the integral of $y\rho$ over the plate is

$\displaystyle \int_0^1 \int_{y^2}^1 xy^2 \, dx \, dy$

and the $y$-coordinate of the center of mass is

$\boxed{\dfrac{\displaystyle \int_0^1 \int_{y^2}^1 xy^2 \, dx \, dy}{\displaystyle \int_0^1 \int_{y^2}^1 xy \, dx \, dy}}$