The radius of an arbitrary shell is its distance from the axis of rotation. If you had rotated about the line x = -1, then the radius would be x - (-1) = x + 1.
I had this hw question (I solved to completion) Let V be the volume of the solid obtained by rotating about the y-axis the region bounded by the following curves.
Find V by cylindrical shell
I understand that the formula is integral 2pi(radius)(height)d(x or y)
To find the height, you do the "top of the shell"-"the bottom of the shell"
in this problem, it was sqrtx-x^2
My question is, why is the radius = to x?
Is it that all cylindrical shell problems that the radius=x? My professor showed us numerous examples and they all had radius=x
I made a note in my book that radius could be the line it touches in a graph
(we had a problem that said to find volume revolving y axis by y=x^2 y=x ... Anyway, after making the cylindrical shell, the radius line we drew hit the line y=x so I figured maybe this was a rule??)
I am confused how to find the radius. Any help appreciated! Thank you.
Since you are rotating about the y axis, the radius is VERTICAL, and so the radius will be the distance from the x-axis to the y-value, so y.
The height of your cylinder is the difference in your x values. We can see that in this region, since , the height will be .
So the surface area of each cylinder will be and so the volume is calculated by