The volume of the solid obtained by rotating the region bounded by the given curves about the specified line and sketch the region, the solid, and a typical disk or washer.
i. If the cross section is a washer with the inner radius rin and outer radius rout, then the area of the 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 solid revolution about y-axis
The region is bounded by x=0,x=9-y2 rotated about the line x=-1
As the region is bounded by x=0,x=9-y2 rotated about the line x=-1,
From the figure, as the region rotates about the line =-1, the strip is perpendicular to y-axis.
A cross section of the solid is the washer with outer radius is 9-y2--1= 10-y2 (distance from x=-1 to x=9-y2) and inner radius is 0--1=1, which is the distance from x=-1 to x=0.
So its area is given by
Ay=πouter radius2-πinner radius2
The region of integration is bounded by x=0,x=9-y2
Therefore, 9-y2=0, y2=9
Here the limits of integration are y=-3 to y=3
Therefore, the volume of the solid revolution about line x=-1,
As the graph of the function is symmetric about x- axis,fx= 99-20y2+y4 is an even function.
By using fundamental theorem of calculus and power rule of integration,
By substituting limits of integration,
The volume of the solid obtained by rotating the region bounded by the given curves