6.3 Integral Applications
Question #4 (Medium): Volume Estimation by Cylindrical Shells Using the Midpoint Rule
Using the cylindrical shells, volume estimation is: ∑ ̅ ̅ , where ̅ is the midpoint
of the -th subinterval[ ].
Given the number of rectangles, the entire interval is divided with a midpoint for each subinterval.
Using the midpoints, the radius and height of each cylindrical shell is computed based on the given
function. Summing all over the subintervals gives the volume estimation.
Estimate the volume using cylindrical shells based on the Midpoint Rule, where and the bound
region is rotated about the given line.
( ), , about
Since and the whole interval is ], then . So the
whole interval is split into ] [ ] [ ] [ ] [ ]