00:01
Hi here for the given question.
00:03
We are given that here we have y square is equal to 4 ax now further here in our case we are given that x is equal to a and a is a positive constant and if the density is directly proportional to the x now here we need to find the center of mass.
00:24
So here center of mass can be written as x bar comma y bar is equal to x comma 1 by 2 times f of x plus g of x.
00:35
So here in our case now further we can say that here the value of mx can be calculated as integration of y dm and my is equal to integration of x dm further here in our case m is equal to integration of dm and x bar can be calculated as my upon m and y bar can be calculated as mx upon m.
01:01
So here in our case now further here we are given that we have our function f of x is equal to plus or minus under root 4 ax which is can also be written as plus or minus 2 times under root ax and here we have line x is equal to a.
01:19
So here y can be written as plus or minus 2 times under root a square which is again equal to y equals to plus or minus 1 upon 2a.
01:29
So here in our case we can say that we have to point a which is a comma minus 2a and point b which is equal to a comma 2a.
01:39
So here in our case we have two function f of x is equal to 2 times under root x and g of x is equal to minus 2 times under root ax.
01:48
So here in our case now further the value of f of x minus g of x is equal to 2 times under root a minus x minus minus 2 times under root a minus x this is equal to 4 times under root ax.
02:04
So here the value of da will be equal to minus 4 under root ax dx.
02:10
So here in our case now further we know that dm is nothing but delta a delta da.
02:18
So this is equal to 4 times delta under root ax dx.
02:23
So here now taking the proportionality constant this is equal to 4kx under root ax.
02:29
So here in our case now further using this value we have m equals to integration over 0 to a 4kx into under root of ax dx.
02:40
So here in our case now further on solving this value we have 4k under root a integration over 0 to a x to the power 3 by 2 dx...