Volume of a bead with square hole- Program approach?

[DM48]

Link to derivation of formula.


\( \int_{0}^{a} \int_{0}^{\sqrt{d^2-y^2-b^2}}\sqrt{d^2-y^2-x^2}-b\text{ }dx\text{ }dy\text{ }+\int_{a}^{d}\int_{0}^{\sqrt{d^2-y^2}}\sqrt{d^2-y^2-x^2}\text{ }dx\text{ }dy \)


In[842]:= Clear[a, b, q, r, x]

d = 12
a = 2
b = 2
q = NIntegrate[
Integrate[
Sqrt[d^2 - y^2 - x^2] - b, {x, 0, Sqrt[d^2 - y^2 - b^2]}], {y, 0,
a}]
r = NIntegrate[
Integrate[Sqrt[d^2 - y^2 - x^2], {x, 0, Sqrt[d^2 - y^2]}], {y, a, d}]
q + r


Out[848]= 857.2260543948448

a=4
b=2
d=12

Out[849]=811.0424359298954