Integration with Cylindrical shells

Oct 2009
5
0
Hello! I have a problem... I don't know how to solve this with the method of cylindrical shells..

The question:
Find the volume of S using the method of cylindrical shells.

S is generated by rotating about the x-axis the region bounded by
y = x^2 , y = 0, and x = 1

(Nerd)
 

HallsofIvy

MHF Helper
Apr 2005
20,249
7,909
Hello! I have a problem... I don't know how to solve this with the method of cylindrical shells..

The question:
Find the volume of S using the method of cylindrical shells.

S is generated by rotating about the x-axis the region bounded by
y = x^2 , y = 0, and x = 1

(Nerd)
Since are rotating around the x-axis, and want to use "cylindical shells", you will want to integrate with resect to y.

The integral will be \(\displaystyle \int_0^1 2\pi r h dy\) where "r" is the radius of the cylinder- that will be just "y", the distance from the x-axis to the line that, rotated around the x-axis, creates the cylinder, and "h" is the length of cylinder, the length of the line that creates the cylinder- that will be "1- x" where x is the x-coordinate of the point (x, y) at the left end of that line: since \(\displaystyle y= x^2\), x= ?
 
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skeeter

MHF Helper
Jun 2008
16,216
6,764
North Texas
Hello! I have a problem... I don't know how to solve this with the method of cylindrical shells..

The question:
Find the volume of S using the method of cylindrical shells.

S is generated by rotating about the x-axis the region bounded by
y = x^2 , y = 0, and x = 1
shells w/r to y ...

\(\displaystyle V = 2\pi \int_0^1 y(1 - \sqrt{y}) \, dy\)


disks w/r to x ...

\(\displaystyle V = \pi \int_0^1 x^4 \, dx\)
 
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