Lecture20 | 1
Variable from to
Representative rectangle (when small)
The area between from to
Lecture20 | 2 7.4. Arclength
Consider a function .
Variable from to
Representative length (when small)
The arc length of from to is
Lecture20 | 3 EXAMPLE. Find the arc length of the graph of
on the interval . ANS
Lecture20 | 4 EXAMPLE. Find the arc length of the graph of
on the interval . ANS
Lecture20 | 5 EXAMPLE. Find the arc length of the graph of
on the interval . ANS
Lecture20 | 6 EXAMPLE. Find the arc length of the graph of
on the interval . Note
Variable from to Arclength
Lecture20 | 7 7.2. Volume: the Disk Method
There are so many solids that are obtained by revolving a certain region about an axis.
Finding the volumes for these objects is another application of integration.
Lecture20 | 8
Lecture20 | 9 Consider the region under the graph
from to . Let’s rotate about the -axis.
Variable from to
Representative disk (when small)
The volume of the solid is
Lecture20| 10 Consider the region between , -axis,
, and . Let’s rotate about -axis.
Variable from to
Representative disk (when small)
The volume of the solid is
Lecture20| 11 EXAMPLE. Find the volume of the solid formed by revolving the region bounded by the graph of
and the -axis ( ) about the -axis.
Lecture20| 12 EXAMPLE. Consider the region bounded by
the graphs of , , , and .
1. Sketch the region .
2. Find the volume of the solid obtained by rotating about the -axis.
Lecture20| 13 EXAMPLE. Find the volume of the solid formed by revolving the region bounded by the graph of
and the -axis, -axis, and about the -axis.
Lecture20| 14
Consider the region between graphs
and .
Lecture20| 15
Variable from to
Representative washer (when small)
The volume of the solid is
Alternatively,
Lecture20| 16 If the axis of rotation is the -axis, the volume of solid is
Lecture20| 17 EXAMPLE. Find the volume of the solid formed by revolving the region bounded by the graphs of
and about the -axis.
Lecture20| 18 EXAMPLE. Find the volume of the solid formed by revolving the region bounded by the graphs of
, , , and about the - axis.
Lecture20| 19 Note If the region: , , ,
lies below the rotation -axis (so ), then
because .
This ambiguity can be removed by noticing that the solid of revolution is the same as that obtained
from , , and , which
lies above the -axis.
Similarly, if the region lies to the left of the rotation -axis, one must reflect to the right, and then apply the formulas we already have.
Lecture20| 20 EXAMPLE. Find the volume of the solid obtained by rotation about the -axis the region given below.
Lecture20| 21 The disk/washer method is used when the solid has circular cross sections:
or
For arbitrary cross section,
Lecture20| 22 EXAMPLE. Find the volume of the solid with
base is the region bounded by the lines
and the cross sections perpendicular to the -axis are equilateral triangles.
Lecture20| 23 7.3. The Shell Method
For the washer method:
Variable from 0 to 4/3
However, it impossible to get and by solving for in terms of !!!
Lecture20| 24
Cylindrical Shells
where is average radius, is the height, is the thickness.
Lecture20| 25
Variable from to
Representative shell (when is small)
The volume of the solid
𝑓(𝑥)
𝑥
Lecture20| 26
Variable from to
Representative shell (when small)
The volume of the solid
𝑓(𝑦)
𝑦
Lecture20| 27 EXAMPLE. Find the volume of the solid obtained by rotating about the -axis the region bounded by
and .
Lecture20| 28 EXAMPLE. Find the volume of the solid obtained by rotating about the -axis the region bounded by
and the -axis ( ).
Lecture20| 29 Generally, if the region is bounded between the
graphs , , , and , and
rotate about -axis,
and if the region is bounded between the graphs
, , , and , and rotate
about the -axis,
Lecture20| 30 EXAMPLE. Use cylindrical shells to find the
volume of the solid obtained by rotating about the -axis the region under the curve from to .
