Yes, I know, it's not clear that we can define an area for a non-orientable surface etc. etc.
So I'll try a more humble question: Following do Carmo, I parameterize the strip by
$ \displaystyle x=(2-v\sin \frac{u}{2})\sin u, y = (2-v\sin \frac{u}{2})\cos u, z=v \cos\frac{u}{2}$
with $0 < u < 2\pi$ and $-1<v<1$.
I now wish to calculate the area of the strip from $0 < u < 2\pi$. This, at least, should be a well defined number and I will leave it to others to decide whether it represents the area of the Möbius strip (or half the area or whatever).
I find
$ \displaystyle \int du dv |x_u \wedge x_v | = \int_0^{2\pi}du\int_{-1}^1dv \sqrt{\frac{v^2}{2}\left(\frac{3}{2}-\cos u\right)-4v\sin\frac{u}{2}+4} $
or, equivalently
$ \displaystyle \int_{-1}^1 dv \int_0^{2\pi} du \sqrt{\frac{v^2}{4}+\left(2-v\sin\frac{u}{2}\right)^2} $
So, first, I've triple-checked this, but it looks awfully complicated for a text book problem. Anyone think I have the wrong expression here? And, finally, anyone have an idea on how to integrate this? I've thought about integration by parts and I've looked up some standard integral forms, but I haven't come up with anything helpful.
EDIT: Just to proof myself from charges of duplication, I know the issue of whether the strip has an area has been discussed on this site. But I have not found a discussion of how to actually do the integral to calculate the area. If someone knows of such a question being asked, then this would be a duplicate. But if not, then it isn't.
EDIT 2: As pointed out in the comment below, do Carmo actually doesn't ask for an evaluation of the integral. But, hey, if anyone can do it, I would still love to see it! Thanks.
A Mobius strip is a two dimensional surface; it has length and width and no thickness.A rectangle is a two dimensional surface; it has length and width and no thickness.Give a rectangle a half twist and join its ends and you get a Mobius strip. The area of both the rectangle and the Mobius strip is equal to length times width.
This is the simplest case. However, Mobius strips can take many different shapes.For example a Sudanese Mobius strip or half a Klein bottle. The area will depend on the particular Mobius strip you are considering.