L ectur e2 0| 1 Volume of solids
Basic solids
a. Circular cylinders and boxes
b. Arbitrary Cylinder
Where is the uniform cross-section area, is the height.
L ectur e2 0| 2 Arbitrary solids
Let be a solid placed along the -axis.
The cross-section area at each is . The volume of is
L ectur e2 0| 3
L ectur e2 0| 4 EX Show that the volume of a sphere of
radius is .
L ectur e2 0| 5 Area of disk/washer
L ectur e2 0| 6 Disk/washer method
Let and a region bounded by Revolve about -axis to get the solid .
The volume of is
L ectur e2 0| 7 Slice (disk)
Radius = Cross-section area =
Volume of slice = Total volume
L ectur e2 0| 8 EX ( -axis) Find the volume of the solid obtained by rotating about the -axis the region under , .
L ectur e2 0| 9 EX Find the volume of the solid obtained by rotating about the -axis for the region enclosed by
L ectur e2 0| 10
Let and a region bounded by Revolve about the -axis to get the solid .
The volume of is
L ectur e2 0| 11
L ectur e2 0| 12 EX ( -axis) Find the volume of the solid obtained by rotating the region bounded by , , , about the -axis.
L ectur e2 0| 13 EX Find the volume of solid obtained by rotating about -axis of the region
enclosed by
L ectur e2 0| 14
EX (Washer) The region enclosed by the curves and is rotated
about -axis. Find the volume of this solid.
L ectur e2 0| 15 Washer method
The volume formula
L ectur e2 0| 16 EX (Arbitrary axis) If the region in
previous example is rotated about the line . Find the volume of the solid.
L ectur e2 0| 17 Shell method
Some problems face difficulty when using disk/washer method.
One cannot solve the equation
for and get and !
L ectur e2 0| 18 Cylindrical Shell
Volume of Shell
where
As , .
L ectur e2 0| 19 Shell method
Let . A region is bounded by , , and . Rotate about -axis.
L ectur e2 0| 20
L ectur e2 0| 21 EX ( -axis) Find the volume of the solid obtained by rotating about the -axis the region bounded by and
.
L ectur e2 0| 22 EX ( -axis) Find the volume of the solid obtained by rotating about the -axis the region between and .
L ectur e2 0| 23 EX ( -axis) Use the cylindrical shells to find the volume of the solid obtained by rotating about the -axis the region under the curve from to 1.
L ectur e2 0| 24 EX Find the volume of the solid obtained by rotating the region bounded by
about the line .
