5
4
3
2
1 y
 x
2 
1
Here is another way we could approach this problem:
0 1 2 cross section
If we take a vertical slice and revolve it about the y-axis we get a cylinder.
If we add all of the cylinders together, we can reconstruct the original object.


5
4
3
2
1 y
 x
2 
1
0 1 2 cross section
The volume of a thin, hollow cylinder is given by:

=2
  
=2
 thickness

2 
 dx r circumference h
r is the x value of the function.
h is the y value of the function.
thickness is dx .


5
4
3
2
1 y
 x
2 
1
This is called the shell method because we use cylindrical shells.
0 1 2 cross section
=2
  
=2
 thickness

2 
 dx r circumference h
If we add all the cylinders from the smallest to the largest:

0
2
2


2 
1
 dx
2
 
0
2 x
3
 x dx 2
   
2



1
4 x
4

1
2 x
2


2
0
12



Find the volume of the solid obtained by rotating the region between y = x 2 +1 and y = 0, x = 0 and x = 1 around the y axis.

Find the volume of the solid obtained by rotating the region between y = x 2 +1 and y = 0, x = 0 and x = 1 around the y axis.
First thing is sketch the graph and think about the rectangles

Find the volume of the solid obtained by rotating the region between y = x 2 +1 and y = 0, x = 0 and x = 1 around the y axis.
First thing is sketch the graph and think about the rectangles
Perpendicular rectangles

Find the volume of the solid obtained by rotating the region between y = x 2 +1 and y = 0, x = 0 and x = 1 around the y axis.
First thing is sketch the graph and think about the rectangles
Perpendicular rectangles

Find the volume of the solid obtained by rotating the region between y = x 2 +1 and y = 0, x = 0 and x = 1 around the y axis.
First thing is sketch the graph and think about the rectangles
Perpendicular rectangles Parallel rectangles

Find the volume of the solid obtained by rotating the region between y = x 2 +1 and y = 0, x = 0 and x = 1 around the y axis.
First thing is sketch the graph and think about the rectangles
Perpendicular rectangles Parallel rectangles

Find the volume of the solid obtained by rotating the region between y = x 2 +1 and y = 0, x = 0 and x = 1 around the y axis.
First thing is sketch the graph and think about the rectangles
Perpendicular rectangles Parallel rectangles
Now think about the methods needed to integrate for each

Find the volume of the solid obtained by rotating the region between y = x 2 +1 and y = 0, x = 0 and x = 1 around the y axis.
First thing is sketch the graph and think about the rectangles
Perpendicular rectangles Parallel rectangles
2 integrals needed
1-Disk [0,1]
1-Washer [1, 2]
Now think about the methods needed to integrate for each

Find the volume of the solid obtained by rotating the region between y = x 2 +1 and y = 0, x = 0 and x = 1 around the y axis.
First thing is sketch the graph and think about the rectangles
Perpendicular rectangles Parallel rectangles
2 integrals needed
1-Disk [0,1]
1-Washer [1, 2]
1 integral needed
Shell Method
*Preferable b/c
(less work) 
Now think about the methods needed to integrate for each

Find the volume of the solid obtained by rotating the region between y = x 2 +1 and y = 0, x = 0 and x = 1 around the y axis.
First thing is sketch the graph and think about the rectangles
Parallel rectangles

Find the volume of the solid obtained by rotating the region between y = x 2 +1 and y = 0, x = 0 and x = 1 around the y axis.
First thing is sketch the graph and think about the rectangles
Parallel rectangles

Find the volume of the solid obtained by rotating the region between y = x 2 +1 and y = 0, x = 0 and x = 1 around the y axis.
First thing is sketch the graph and think about the rectangles
Parallel rectangles

Find the volume of the solid obtained by rotating the region between y = x 2 +1 and y = 0, x = 0 and x = 1 around the y axis.
First thing is sketch the graph and think about the rectangles
Parallel rectangles

Find the volume of the solid obtained by rotating the region between y = x 2 +1 and y = 0, x = 0 and x = 1 around the y axis.
First thing is sketch the graph and think about the rectangles
Parallel rectangles

Find the volume of the solid obtained by rotating the region between y = x 3 + x + 1, y = 1 and x = 1 around the line x = 2.

Find the volume of the solid obtained by rotating the region between y = x 3 + x + 1, y = 1 and x = 1 around the line x = 2.
Immediately we should notice y=x 3 + x + 1 is not easily solvable for x so washer method is out!
Shell Method is NECESSARY!

Find the volume of the solid obtained by rotating the region between y = x 3 + x + 1, y = 1 and x = 1 around the line x = 2.
Immediately we should notice y=x 3 + x + 1 is not easily solvable for x so washer method is out!
Shell Method is NECESSARY!

Find the volume of the solid obtained by rotating the region between y = x 3 + x + 1, y = 1 and x = 1 around the line x = 2.
Immediately we should notice y=x 3 + x + 1 is not easily solvable for x so washer method is out!
Shell Method is NECESSARY!

Find the volume of the solid obtained by rotating the region between y = x 3 + x + 1, y = 1 and x = 1 around the line x = 2.
Immediately we should notice y=x 3 + x + 1 is not easily solvable for x so washer method is out!
Shell Method is NECESSARY!

Find the volume of the solid obtained by rotating the region between y = x 3 + x + 1, y = 1 and x = 1 around the line x = 2.
Immediately we should notice y=x 3 + x + 1 is not easily solvable for x so washer method is out!
Shell Method is NECESSARY!

Find the volume of the solid obtained by rotating the region between y = x 3 + x + 1, y = 1 and x = 1 around the line x = 2.
Immediately we should notice y=x 3 + x + 1 is not easily solvable for x so washer method is out!
Shell Method is NECESSARY!

Find the volume of the solid obtained by rotating the region between y = x 3 + x + 1, y = 1 and x = 1 around the line x = 2.
Immediately we should notice y=x 3 + x + 1 is not easily solvable for x so washer method is out!
Shell Method is NECESSARY!

Find the volume of the solid obtained by rotating the region between y = x 3 + x + 1, y = 1 and x = 1 around the line x = 2.
Immediately we should notice y=x 3 + x + 1 is not easily solvable for x so washer method is out!
Shell Method is NECESSARY!

Find the volume of the solid obtained by rotating the region between y = x 3 + x + 1, y = 1 and x = 1 around the line x = 2.
Immediately we should notice y=x 3 + x + 1 is not easily solvable for x so washer method is out!
Shell Method is NECESSARY!

Find the volume of the solid obtained by rotating the region between y = x 3 + x + 1, y = 1 and x = 1 around the line x = 2.
Immediately we should notice y=x 3 + x + 1 is not easily solvable for x so washer method is out!
Shell Method is NECESSARY!