Area by Parts and Rotational Volume
Shell Method
The shell method: The concept is adding an infinite number of shells together to find area.
For rotation around the y axis the formula is: 
For example:
4.1 Let R denote the region bounded by the x axis, the function y = x² and the line x = 3.
Find the volume of R when rotated around the y axis.
For a rotation around the x axis the formula is: 
For example:
4.2 Let R denote the region bounded by the y axis, the function 
.
Find the volume of R when rotated around the x axis.
To find the integral’s boundaries find y when x is equal to 0.
Rotating around a line rather than an axis with shell:
The radius changes from either x or y to (the line – x) or (the line – y).
For example:
4.3 Let R denote the region bounded by 
the y axis and the x axis.
Find the volume of R rotated about the line 
(when x ≥ 0)
To find the integral’s boundaries find x when y is equal to 0.
but throw away the negative 2 because it is less than 0.
because it is rotated around , the radius is 
rather than x
