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Find center of mass of region bounded by `y= sinx`  between  `x=0` and `x=pi`  above...

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pensivegal | (Level 1) Honors

Posted October 3, 2013 at 9:39 PM via web

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Find center of mass of region bounded by `y= sinx`  between  `x=0` and `x=pi`  above x-axis. 

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aruv | High School Teacher | (Level 2) Valedictorian

Posted October 4, 2013 at 4:04 AM (Answer #1)

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Area of the region between curves y=sin(x) and y=0 bounded by x=0 and x=pi

`Delta=int_0^pi(sin(x)-0)dx`

`=-(cos(x))_0^pi`

`=-[cos(pi)-cos(0)]`

`=2`

Let coordinate of centre of mass be `(barx,bary).`  Then

`barx=(1/2)int_0^pix sin(x)dx`

`=(1/2){-xcos(x)+sin(x)}_0^pi`

`=pi/2`

`bary=(1/2)int_0^pi(1/2)sin^2(x)dx`

`=(1/8)int_0^pi(1-cos(2x))dx`

`=(1/8){x-sin(2x)/2}_0^pi`

`=pi/8`

Thus mass centre is `(pi/2,pi/8).`

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