Timeline for Integral of the square of the areas of slices of a shape
Current License: CC BY-SA 4.0
7 events
| when toggle format | what | by | license | comment | |
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| May 5, 2018 at 6:16 | comment | added | Ben McKay | It is clear from the case of a rectangle that this integral is not rotation invariant, so it doesn't compute a geometric quantity. | |
| May 4, 2018 at 17:44 | review | Close votes | |||
| May 5, 2018 at 10:04 | |||||
| May 4, 2018 at 17:36 | comment | added | Peter LeFanu Lumsdaine | @BeniBogosei: Saying that “in the case where the integrand is $A_t$, the integral is equal to the volume” seems more of a geometric interpretation than a way to explicitly compute the integral? If you want an integral over the whole body recovering your integral-of-squares-of-slices, then you can take $\int_{(x,y,z) \in \omega} A_z dx dy dz$; this again doesn’t seem to me like it helps to “explicitly compute” the original, but I still don’t follow what kind of formula you’re hoping for, so maybe it is of use? | |
| May 4, 2018 at 15:14 | comment | added | Beni Bogosel | @PeterLeFanuLumsdaine: In the case where the integrand is $A_t$, the integral is equal to the volume. For me, a satisfactory answer would be that the integral can be computed by integrating a function on the whole body $\omega$, rather than integrating with respect to the height parameter. I'm not sure such an answer exists. At least I didn't find one, but before deciding that this isn't possible, I wanted to ask the question. | |
| May 4, 2018 at 13:43 | comment | added | Peter LeFanu Lumsdaine | It’s not clear to me what sort of formula you’re looking for — for general $\omega$, what kind of formula could one hope for that would be more explicit than $\int_0^T (A_t)^2 dt$? Can you give an example to illustrate what you’re looking for — i.e. a formula to “explicitly compute” some other integral associated to an object, in the sense you have in mind? | |
| May 4, 2018 at 12:11 | history | edited | Piero D'Ancona | CC BY-SA 4.0 |
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| May 3, 2018 at 22:25 | history | asked | Beni Bogosel | CC BY-SA 4.0 |