>>528
解析解は諦めてモンテカルロで概算をだしてみる。プログラムも全然楽だったY

WineGlass <- function(deg=0,N=1e7){ # deg degree tilted
wine <- function(x,z) -cos(sqrt(x^2+z^2)) # rotation of y=-cos(x) around y-axis
x=runif(N,-pi,pi)
y=runif(N,-1,1)
z=runif(N,-pi,pi)
θ=deg*pi/180
x0=pi-asin(tan(θ)) # x-cordinate of the edge of filled wine in tilled glass
y0=-cos(x0) 　 # y-cordinate
(2*pi)*2*(2*pi)*mean(y >wine(x,z) & x^2+z^2<pi^2 & y<tan(θ)*(x-x0)+y0)
}
WineGlass=Vectorize(WineGlass,vectorize.args='deg')
y=WineGlass(0:20)

data.frame(deg=0:20,volume=y,ratio=y/y[1])
> data.frame(deg=0:20,volume=y,ratio=y/y[1])
deg volume ratio
1 0 18.43 1.000
2 1 17.01 0.923
3 2 15.78 0.856
4 3 14.67 0.796
5 4 13.67 0.742
6 5 12.73 0.691
7 6 11.87 0.644
8 7 11.04 0.599
9 8 10.29 0.559
10 9 9.58 0.520
11 10 8.90 0.483
12 11 8.25 0.448
13 12 7.66 0.416
14 13 7.09 0.385
15 14 6.55 0.355
16 15 6.04 0.328
17 16 5.58 0.303
18 17 5.11 0.278
19 18 4.69 0.255
20 19 4.28 0.232
21 20 3.92 0.213

９〜１０°傾ければいいみたい。