T sphere_relative_area(const std::type_identity_t<T> a, const std::type_identity_t<T> b)

{

static_assert(std::is_floating_point_v<T>);

// Assuming[Element[n,Integers]&&n>=0,Integrate[Sin[x]^n,x]]

// -Cos[x] Hypergeometric2F1[1/2,(1-n)/2,3/2,Cos[x]^2] Sin[x]^(1+n) (Sin[x]^2)^(1/2 (-1-n))

// For[i=2,i<=10,++i,f=Integrate[Sin[x]^(i-2),{x, a, b}];Print[i];Print[f]]

// For[i=2,i<=10,++i,f=Simplify[Integrate[Sin[x]^(i-2),x]];Print[i];Print[f]]

{

return b - a;

return std::cos(a) - std::cos(b);

return (2 * b - 2 * a - std::sin(2 * b) + std::sin(2 * a)) / 4;

return (9 * std::cos(a) - std::cos(3 * a) - 9 * std::cos(b) + std::cos(3 * b)) / 12;

return (-12 * a + 12 * b + 8 * std::sin(2 * a) - std::sin(4 * a) - 8 * std::sin(2 * b)

+ std::sin(4 * b))

/ 32;

return (150 * std::cos(a) - 25 * std::cos(3 * a) + 3 * std::cos(5 * a) - 150 * std::cos(b)

+ 25 * std::cos(3 * b) - 3 * std::cos(5 * b))

/ 240;

return (-60 * a + 60 * b + 45 * std::sin(2 * a) - 9 * std::sin(4 * a) + std::sin(6 * a)

- 45 * std::sin(2 * b) + 9 * std::sin(4 * b) - std::sin(6 * b))

/ 192;

return (1225 * std::cos(a) - 245 * std::cos(3 * a) + 49 * std::cos(5 * a) - 5 * std::cos(7 * a)

- 1225 * std::cos(b) + 245 * std::cos(3 * b) - 49 * std::cos(5 * b) + 5 * std::cos(7 * b))

/ 2240;

return (-840 * a + 840 * b + 672 * std::sin(2 * a) - 168 * std::sin(4 * a) + 32 * std::sin(6 * a)

- 3 * std::sin(8 * a) - 672 * std::sin(2 * b) + 168 * std::sin(4 * b) - 32 * std::sin(6 * b)

+ 3 * std::sin(8 * b))

/ 3072;

return ns::numerical::integrate<T>(

[](const T x)

{

return std::pow(std::sin(x), static_cast<T>(N - 2));

},

a, b, /*count=*/100);

}

}

}