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{(\displaystyle \int_0^1 \int_(y^2)^1 xy^2 \, dx \, dy)/(\displaystyle \int_0^1 \int_(y^2)^1 xy \, dx \, dy)}`