= Solution
For the <Komar integral of a five-dimensional Reissner--Nordstrom metric>, use the one-form metrically dual to the stationary <Killing vector field>: $k^\flat=-f(r)dt$. Its <exterior derivative> is $dk^\flat=f'(r)dt\wedge dr$. The stated orientation gives volume form
$$
\epsilon=-r^3\sin^2\chi\sin\theta\,dt\wedge dr\wedge d\chi\wedge d\theta\wedge d\phi.
$$
Because the squared norm of $dt\wedge dr$ is $g^{tt}g^{rr}=-1$, the defining wedge identity for the <Hodge star operator> gives
$$
\star(dt\wedge dr)=r^3\sin^2\chi\sin\theta\,d\chi\wedge d\theta\wedge d\phi.
$$
Thus the <Komar integral> over the positively oriented angular three-sphere is
$$
\int_{S^3}\star dk^\flat=2\pi^2r^3f'(r),
$$
since the unit three-sphere has volume $2\pi^2$. Write $s=r_+^2+r_-^2$, $q=r_+^2r_-^2$. Then $f=1-s/r^2+q/r^4$ and $r^3f'=2s-4q/r^2$. With the normalization specified for the <Komar mass>,
$$
\boxed{M=\frac\pi4(r_+^2+r_-^2).}
$$
The orientation makes this positive. The problem's coefficient $1/(16\pi)$ must be retained: it is not the usual dimension-adjusted five-dimensional ADM-normalized Komar coefficient. With that conventional normalization the ADM <mass> would instead be $3\pi s/8$.
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