6.2 Integral Applications
Question #2 (Medium): Finding the Volume of a Solid Rotated About a Vertical Line
When the area is rotated about a vertical line, the extends horizontally, then naturally the
of the disks will rise vertically, yieldinor . Therefore, integration is done with respect to
and all functions in the integral are written in terms of . If the functions are initially written in terms of
, they must be rearranged to be re-written in terms of . The volume is then:
∫ , where the is expressed in terms of . Whether integrate over or ,
the area of the cross-sectional disk is still evaluated as: .
The interval[ ]is the start and ending point along the -axis.
Find the volume of the solid that is obtained by rotating the area bound by the functions about the given
line. Include the sketch of the bound area, as well as the solid and a typical sample disk or washer.
, , rotated about the -axis
The graph of the functions is as follows:
Rotation about the -axis means rotation about a vertical line. So the