Please help Find the center of mass

Question

Calculus II



1- Find the center of mass of the lamina of constant density occupying the region between

y=4-2x and y=2x in the first quadrant.

2- Find the center of mass of the lamina of uniform density occupying the region under y=x^3 for

0 ≤ x ≤ 2

Answer 1

1) Looking at the graph shows that the curves form a triangle in the first quadrant. They intersect at at (1,2) with the base going from 0 to 2. So the center of mass is the center of area which is 1/3 up the triangle which would be (1/3)*2 for y and by symmetry, x = 1.

So the center of area = center of mass = (1,2/3).

2) This requires the center of area integrals for the center of area which is the center of mass in this case.

Solving for xbar and ybar, we find

xbar = 1.6 and ybar = 2.286