The volume of the solid obtained by rotating the region bounded by the given curves about the line y= -3 and sketch the region, the solid, and a typical disk or washer.
The sketch of the bounded region, the solid, and a typical washer:
i. If the cross section is a washer with the inner radius rin and the 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 the solid revolution about x-axis is
y=x3, y=1,x=2;line y=-3
The region bounded by y=x3, y=1,x=2 and solid of revolution about the line y=-3 are shown below.
Here, the region is rotated about the line y=-3, so the cross-section is perpendicular to the x-axis.A cross section of the solid is the washer with the outer radius x3--3=x3+3 and the inner radius 1--3=4.
So, its cross sectional area is
Ax=πouter radius2-πinner radius2 =πx3+32-π42
The region of integration is bounded by y=x3, y=1 and x=2.
At intersection of y=x3, y=1,
The other boundary is x=2, sos solid lies between x=1 and x= 2
Therefore, the volume of the solid of revolution about the line y=-3,
By using the fundamental theorem of calculus part 2 and the power rule of integration,
By substituting the limits of integration,
Therefore, the volume of the solid obtained by rotating the region bounded by the given curves about the line y=-3 is V=471π14
The volume of the solid obtained by rotating the region bounded by the given curves about the line y=-3 is V=471π14