Numerical integral or Numerical value of a symbolic integral

Question

I'm new to numerical integration. Why is this code wrong?

N[Integrate[(b^2 Sin[θ]^3 Cos[ϕ]^2)/(b^2 Cos[ϕ]^2  Sin[θ]^2 + a^2 Sin[ϕ]^2 Sin[θ]^2 +
 a^2 b^2 Cos[θ]^2), {θ, 0, Pi}, {ϕ, 0, 2 Pi}]]

$a$ and $b$ are real values.

Answer 1

You can give Integrate assumptions, e.g.:

integrand = (b^2 Sin[θ]^3 Cos[ϕ]^2)/(b^2 Cos[ϕ]^2 Sin[θ]^2+a^2 Sin[ϕ]^2 Sin[θ]^2+a^2 b^2 Cos[θ]^2);

Integrate[
    integrand,
    {θ,0,π},
    {ϕ,0,2 π},
    Assumptions->(a|b) ∈ Reals
]

$Aborted

However, I wasn't patient enough to let the above finish. An alternate method (e.g., as mentioned here) is to substitute real constants for the parameters, and then substitute back. For example:

f[a_, b_] = Integrate[
    integrand /. {a->EulerGamma, b->Khinchin},
    {θ, 0, π},
    {ϕ, 0, 2 π}
] /. {EulerGamma->a, Khinchin->b}; //AbsoluteTiming

f[a, b] //TeXForm

{17.7305, Null}

$\frac{4 \pi b \left(\frac{a \sqrt{b^2-a^2} \left(E\left(\frac{1-b^2}{a^2-b^2}\right)-E\left(\sin ^{-1}(a)|\frac{1-b^2}{a^2-b^2}\right)\right)}{a^2-1}+b\right)}{b^2-a^2}$

When using this approach it is best to compare the numerical answer with the proposed answer:

f[1/4, 1/3] //N
f[11/4, 2/3] //N

6.58407 + 0. I

1.16785 + 0. I

Versus:

g = With[
    {i = integrand /. {a->#1, b->#2}},
    Function[NIntegrate[i, {θ, 0, π}, {ϕ, 0, 2 π}]]
];

g[1/4, 1/3]
g[11/4, 2/3]

6.58407

1.16785