Center of mass for a function defined body

[Redoctober]

Homework Statement

The upper side of a uniform plate of thickness w is given by a function y(x) = 4a+x^2/a. Length of this plate on the x-axis is a . Find x coord of the center of mass of this object, with respect to the origin O . Take density of the plate as p

The Attempt at a Solution

Y(x)= 4a+(x^(2))/a

Can be restated as w.dA=w(4a+(x^(2))/a).dx
therefore using
w.dA.p=dm
w(4a+(x^(2))/a).dx.p=dm

distance from O to CoM qouted as S
will be S= 1/M∫x.dm
S=1/M∫x.w(4a+(x^(2))/a).p.dx
S=1/M.w.p∫(4ax+(x^3)/a).dx

therefore finally , integrating from x=0 to x=a

S=1/M.w.p(2a^3+(a^3)/4)
Mass M can be substituted by = p.w.∫1.dA = p.w.∫y(x).dx from 0 to a

Is this is correct ?! :) Thanks in advance

:)

 

[Spinnor]

You have roughly a rectangle of width a and height 4a. The center of mass you get should be close to (a/2,2a)? Did you get an answer close to this?

 

[Redoctober]

I did it for the x distance from the origin , i got 27/52*a which is about o.519a i.e nearly 1/2a

 

[ehild]

I did it for the x distance from the origin , i got 27/52*a which is about o.519a i.e nearly 1/2a

it is correct.

ehild