Shell/disk method help:( Can someone please help me with this? I do not understand how to get the bounds and do the problem. please give an example and explanation.
1 Answer
See below.
Explanation:
This is for the disc method.
From the diagram we can see that the radius is the distance from
.i.e.
The radius to the line
Each disc for
Each disc for
We need to subtract the volume generated by
Our bounds are
Volume = V.
V
V
V
Plugging in bounds:
V
V
This is for the shell method:
In this method instead of using disc perpendicular to the axis of rotation, we use cylinders or shells these can be expanded or contracted to completely cover the solid we are seeking. In this method the width of the shell is not the function value as the radius was in the disc method. Here the width of the shell is the distance from the
Since we are working in relation to the y axis we need our function to be terms of
i.e.
The distance out to the edge of the shell is
The cross sectional area is:
The first shell will enter the solid at
The volume of the solid will then be:
Exactly the same method is used with
So our combined integral will be:
V
Plugging in bounds:
V
V
So both methods yield the same result.
Volume of revolution:
Hope this helps:
As you can see the shell method isn't as intuitive as the disc method. This method takes a fair amount of study to get comfortable with. The main use of this method is when it isn't possible to set up a disc method. The main time you need this is when revolving around a vertical line