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.
I need these please: m, Mx, My So I can show the work. I just don\'t know how to set them up.
The density is given by p(x, y)=3y. Find the center of mass.
I need these please: m, Mx, My So I can show the work. I just don\'t know how to set them up.
Solution
m = double integral (3y)dydx limits y from 0 to sinx and x from 0 to pi m = double integral (3y)dydx = integral ( 3*(sinx)^2/2)dx = (3/4)* integral( 1 - cos2x ) dx = (3/4)* [ (pi - 0) - (cos2pi - cos0)/2] = 3pi/4 Mx = pi/2 ( you can easily say this from symmetry of lamina and independence of density of x) ( still i\'ll write the formula for you Mx = integral ( y* inegral(3y)dy )dx / m limits : y from 0 to sinx x from 0 to pi ) My = integral ( y* inegral(3y)dx )dy / m limits : x from arcsin y to arcsin y + pi/2 y from 0 to 1 My = integral (y*(3y*pi/2))dy/m = (3pi/2m) integral(y^2)dy = (1/2) * ( 1/3 ) = 1/6
