When 3x2=3√x at the points of intersection.
x2=√x⇒x=0,1 and respectively y=0,3, that is, (0,0) and (1,3) are the intersection points.
In the shell method the x-axis is the axis of rotation so y forms the radius of the cylindrical shells and x forms the height. Each shell is a cylinder lying on its side. The height varies, starting at zero height at (0,0), then growing and shrinking back to zero at (1,3). To find the height we need x in terms of y:
x2=y/3 so x=√(y/3) and √x=y/3, so x=y2/9. The height is √(y/3)-y2/9, which is positive between y=0 and y=3; for example, at y=¾, the height is ½-1/16=7/16.
The volume of each cylindrical shell is 2πy(√(y/3)-y2/9)dy, and the sum of all the infinitesimal shells is:
2π0∫3y(√(y/3)-y2/9)dy=
2π0∫3(y3/2/√3-y3/9)dy=
2π[⅖y5/2/√3-y4/36]03=2π(18/5-9/4)=2.7π cubic units.