Question

Find the mass and center of mass of the lamina that occupies the region D and has the given density function ρ.
D is bounded by
y =sqrt(x), y = 0, and x = 1; ρ(x, y) = 27x

m=?
(x,y)=?

Answer 1

consider the lamina as a collection of thin vertical strips, of height y and width dx. Just add up all the masses.

Each strip's mass is its area times it density. If we integrate along x, the density is a constant for each strip.

m = ∫[0,1] ρ y dx
= ∫[0,1] 27x √x dx
= 54/5 x^(5/2) [0,1]
= 54/5