Who is good at calculus? finding volumes by cylindrical shells

[03cobra 87GN]

hey guys i was working on a problem from my class and I'm having some problems with it. I keep getting a neg. answer which it can't be since I'm trying to find a volume of a function rotated around the y-axis

I'm posting this here hopefully i can get some pretty fast since i know alot of people are on this site.

here is a link to another forum that uses syntax to make it easier to see
problem with volume using shells : Calculus

the function being rotated is y=1/\sqrt{1-x^2}
boundaries are y=0 lower bound is x=0 and upper is x= sqrt{2}/2


the formula i used is y=2*\pi *r*h*w

r= radius and is x since function is rotated about y-axis
h=height and is Y(top)-Y(bottom) function is on top so it is just the function

so the set up is delta vol.=2*pi *x*(1/sqrt{1-x^2})delta x

so to integrate I need to do a U substitution my U I used is U=1-x^2

so du/dx=-2x

than subbing du in the integral becomes -\pi (1/sqrt{u})du
next i want to integrate with the u so than i would need to change the bounds
lower x=0 u=1-x^2 so u=1

upper x=sqrt{2}/2 u=1-x^2 so u=1/2

so my integral now looks like vol. = -pi int_{1}^{1/2}(1/sqrt{u})du


than integrating that i got 2*sqrt{u} so i can pull the two through integral
vol.= -2*pi \int_{1}^{1/2}(sqrt{u})du

now i want to solve it so i replace u with u=1-x^2 in intergral

this is where I'm having problems. i keep getting a neg. answer

what is have so far is vol.=-2*pi [ (sqrt{1-(1/2)^2})-(sqrt{1-1^2}) ]

vol.=-2*pi [\sqrt{1-(1/4)}-(\sqrt{1-1}) ]
-2*pi[ sqrt{3}/2-0 ]

thank you for any help,
-Tim

 

[UFGatorGuy20]

You set up everything correctly. Your formula and bounds are correct. I think you just applied your limits wrong after integrating. Remember... you converted to the U domain, so your bounds must be correct as well. After you integrate in U, apply your limits in U. Don't plug X back in. You plugged in U = 1 - X(squared), but still applied limits in terms of U and not X.

I think...

 

[03cobra 87GN]

yah I noticed that also idk why I did that:shrug: thanks for helping.
-Tim

 

[F the V-TAC]

Ahhh...nothing like a little shells method. Finally past that chapter in the year