The solid for the integral π∫01y4-y8dy which represents the volume of a solid
= x, y/0≤y≤1,y4≤x≤y2, about the y- axis
i. If the cross section is a washer with inner radius rin and outer radius rout, then area of washer is obtained by subtracting the area of the inner disk from the area of the outer disk,
A=π outer radius2 -π inner radius2
ii. The volume of the solid revolution about y-axis is
Comparing the below integral with expression for volume,
We see that the cross section of the solid of which the above integral gives volume, is a washer with outer radius y2 and inner radius y4 and cross section is perpendicular to the axis of rotation y axis.
So, the solid is obtained by rotating the area bounded by the curve x= y4 and the curve x= y2 and it lies between y=0 and y=1 about y-axis.
This can be written as
Therefore, the givenintegral describes the volume of the solid obtained by rotating the region
= x, y/0≤y≤1,y4≤x≤y2 of the xy- plane about the y- axis
Conclusion: The integral
describes the volume of the solid obtained by rotating the region