Homework SourseA lamina R is bounded by the xaxis and y sin x the first arc
A lamina ,R, is bounded by the x-axis and y =sin x (the first arch in quadrant I).
The density is given by p(x, y)=3y. Find the center of mass.
The density is given by p(x, y)=3y. Find the center of mass.
Solution
As far as the x coordinate is concerned, because of symmetry, the center of mass is at x = /2.
Now take small element dx.
Center of mass of this dx is at (x, y/2).
As dx is infinitesimally small, it can be thought of as a rectangle with area = y * dx.
Now for finding out yCM we need the total area.
A = Integral of y dx from 0 to = Integral of sin x dx from 0 to = 2
Now yCM = (integral of (y/2)dA from 0 to )/A = (integral of (sin x)2dx) / 4
= integral of (1 - cos(2x))dx / 8 = /8
Thus (/2, /8) is the answer.
