Define the function for all
. Let
Compute
- A(t);
- V(t);
- W(t);
.
- The area of the ordinate set on
is given by the integral,
- The volume of the solid of revolution obtained by rotating
about the
-axis is
- To compute the volume of the solid of revolution obtained by rotating
about the
-axis we first find
as a function of
.
Since
, the integral is then from
to 1 and we have
- Finally, using parts (c) and (d) we can compute the limit,
The solution (c) is incorrect: you need to use the integral of
. This will not directly take one to the ending answer. After the integral is computed (can be easily done with integration by parts), you have to add
to the result – this is the portion of the solid of revolution, that lies below the
, it is a square, which revolves around the y-axis. This will give the answer in Apostol. If sth, one can plot this function and immediately see this.
In part c you forgot to square the function . It should be integral of pi x^2
(I think)
This part of the problem was actually meant to get us to use the method of cylinder “shells” in integrating the solid (Circumference * Height, Integrated over dx).
The integral should look like this (I hope my formatting works…):
When calculating the integral using integration by parts, I got the correct, back-of-book answer.
Credit to the posters at Stack Exchange for reminding me that the “shell” method is a thing.
https://math.stackexchange.com/questions/934772/problem-with-volume-of-solid-of-revolution