Find the centroid of the region bounded by the given curves. y=x^{3},x+y=2,y=

Concepcion Hale 2021-12-10 Answered
Find the centroid of the region bounded by the given curves. y=x3,x+y=2,y=0
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Expert Answer

Linda Birchfield
Answered 2021-12-11 Author has 39 answers
f(x)=2x, g(x)=x3
or x=2y, x=y13
They intersect at (1,1)
Since integrating with respect to x would mean we need to do two separate integrals for everything (from x=0 to l and from x=1 to 2), we could alternatively integrate with respect to y, where x=2 — y is the "top" function and x=y13 is the "bottom",

Find the area of the bounded region.
A=01((2y)y13)dy
=[2y12y234y43]01
=21234(000)
=8234
=34
If we change all the x to y in the formula to find x, it will give the y coordinate of the centroid.
y=13401y((2y)y13)dy=4301(2yy2y43)dy
=43[y213y337y73]01
=43[11337(000)]
=43[217921]
=43[521]
=2063
Likewise, if we change all the x to y in the formula to find y, it will give the x coordinate of the centroid.
lalilulelo2k3eq
Answered 2021-12-12 Author has 38 answers
Say f(x) and g(x) are the two bounding functions over [a,b]
The mass is M=abf(x)g(x)dx
We find the moments:
Mx=12ab([f(x)]2[g(x)]2)dx
My=abx(f(x)g(x))dx
And the center of mass, (x,y), is
x=MyM
y=MxM
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