The referenced post is 

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[__this one__](https://www.reddit.com/r/Geometry/s/HAWUmKip8n) ,

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but I've shown the figure the volume of which is being queried _anyway, here_ , as the frontispiece.

So I came up with an expression that's precise to product of _two_  of the small quantities - the __'small quantities'__ being the radius (say __Q__) of the rounding of the upper edge (assuming it to be circular), & the thickness (say __H__). Also let the radius be __R__ ; & also let the distance of the chord constituting the _upper_  straight edge from the diameter to which it's parallel be __X__ ; & let the angle the slope _from_  that upper edge makes to the vertical be __Θ__ . So __H__ & __Q__ can fairly reasonably be dempt to be small fractions of __R__ , whereas __X__ is a substantial fraction of __R__ & needs to be treated as a quantity of the same order of size as __R__ . Then the expression I came up with for, as I said above, the volume precise to product of _two_  of the small quantities is

__2H(X+HtanΘ)√(R^(2)-X^(2))__

__+ R(2RH-(4-π)Q^(2))arcsin(X/R)__ .

I'm _fairly_  sure that's correct ... but let it be part of this query whether I've made an error with that.

&nbsp;

But it kept pecking @ me whether a fully precise expression couldn't be derived (I mean, _ultimately_  it obviously _can be_  derived) ... & I came up with the idea that the best way to calculate the volume is to integrate _along the axis of_  the underlying thick disc - or squat cylinder - that the figure is extracted from by cutting parts away ... & I came-up with the following expression. 

__Volume = 2×(__

__∫{0≤z≤H-Q}(__

__(X+(H-z)tanΘ)√(R²-(X+(H-z)tanΘ)²)__

__+__

__arcsin((X+(H-z)tanΘ)/R)__

__)dz__

__+__

__∫{0≤z≤Q}(__

__(X+(Q-z)tanΘ)√((R-Q+√(Q²-z²))²-(X+(Q-z)tanΘ)²)__

__+__

__arcsin((X+(Q-z)tanΘ)/(R-Q+√(Q²-z²)))__

__)dz__

__)__ .

The integral is that because @ each __z__ the crosssection the area of which is to be integrated _with respect to_  __z__ is a disc of radius __r__ that has two regions, each between a chord @ given distance __x__ from the diameter & the edge of the disc (& @ opposite sides of it), removed. And it's a standard result that that area is

__2(x√(r²-x²)+r²arcsin(x/r))__ .

And upto where the rounding of the upper edge begins - ie @ distance __Q__ before the upper limit __H__ - __x__ varies & is given by

__X+(H-z)tanΘ__

& __r__ is constant; but _into_  the region beyond that both __x__ _and_  __r__ vary, with (& with __z__ now being distance into the region with the rounded edge, or _the original_  __z__ less __H-Q__)

__x = X+(Q-z)tanΘ__ &

__r = R-Q+√(Q²-z²)__ .

So that's the best solution I've got so-far - I'm fairly sure that integrating along the __z__ -axis (ie along the axis of the underlying cylinder) _is_  the simplest way of doing it: other ways of slicing it I tried resulted in integrals that were not-only _of nested radicals_ , but ___double___ integrals, also! ... but I'm not _absolutely sure there isn't_  a better one (but let _that_  be part of the query, also). And it will be noticed that the second part of the integral - ie the part that applies where there is the rounding of the edge of the squat cylinder - entails nested radicals.

Now looking-up integrals of nested radicals, I find there seems to be _prettymuch nothing_ , treatise-wise, online about it. I found a few items about integrals of _particular infinitely_ -nested radicals ... but nothing dealing with integration of nested radicals _in-general_  - including both infinitely-nested _and_  finitely-nested ones. So I don't know whether there's a closed form expression for the one occuring here. (And _even if there is_ it's doubtful whether that one the integrand of which is the __arcsin()__ of a complex-ish function wouldn't require numerical integration _anyway_ .)

So I wonder whether anyone can either adduce a closed-form expression for the integral of nested radicals that appears here (and maybe for that __arcsin()__ of complex-ish function one, aswell, even ... which would actually completely solve the original problem @ r/Geometry in-terms of closed-form expressions); or signpost to somekind of treatise on integration of nested radicals.

Arisal of Integral Entailing Nested Radicals in the Calculation of the Volume of a Certain Figure Specified in a Recent Post in the Geometry Subreddit