Centroid of volume formed by rotating about the y-axis the area bounded by 3 lines

Centroid of the solid generated by the area Bounded by y=x, y=3x and X + Y= 8 about y axis

Using Vertical Strip
V=Σ[2πx2x1xydx]

V=2π20x(3xx)dx+2π42x[(8x)x]dx

V=32π unit3
 

integral_011-volume-revolved.gif

 

By symmetry
XG=0
 

Solving for YG
VYG=Σ[2πx2x1ycxydx]

VYG=Σ[2πx2x112(yU+yL)x(yUyL)dx]

VYG=Σ[πx2x1x(yU2yL2)dx]

32πYG=π20x(9x2x2)dx+π42x[(8x)2x2]dx

32πYG=3523π

YG=113
 

Centroid of the solid is at (0, 11/3)
 

Using Horizontal Strip
V=Σ[πx2x1(xR2xL2)dy]

V=π40(y219y2)dy+π64[(8y)219y2)]dy

V=32π unit3
 

integral_011-volume-revolved-hor.gif

 

VYG=Σ[πx2x1yc(xR2xL2)dy]

32πYG=π40y(y219y2)dy+π64y[(8y)219y2)]dy

32πYG=3523π

YG=113
 

Centroid of the solid is at (0, 11/3)