Find the centroid of the region bounded by the given curves. y=x3,x+y=2,y=0

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Linda Birchfield · Accepted Answer

f(x)=2−x, g(x)=x3  or x=2−y, 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 &quot;top&quot; function and x=y13 is the &quot;bottom&quot;,    Find the area of the bounded region.  A=∫01((2−y)−y13)dy  =[2y−12y2−34y43]01  =2−12−34−(0−0−0)  =8−2−34  =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―=134∫01y((2−y)−y13)dy=43∫01(2y−y2−y43)dy  =43[y2−13y3−37y73]01  =43[1−13−37−(0−0−0)]  =43[21−7−921]  =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 · Accepted Answer

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=12∫ab([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

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

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