= Solution
In the first octant, <spherical polar coordinates> have $0\leq\theta\leq\pi/2$ and $0\leq\phi\leq\pi/2$. The <charge density> is $\lambda xy=\lambda r^2\sin^2\theta\cos\phi\sin\phi$. Hence the total <electric charge> is
$$
Q=\lambda\int_0^a r^4\,dr\int_0^{\pi/2}\sin^3\theta\,d\theta\int_0^{\pi/2}\cos\phi\sin\phi\,d\phi.
$$
The three factors are $a^5/5$, $2/3$, and $1/2$, respectively. For the middle <integral>, set $u=\cos\theta$ to obtain $\int_0^1(1-u^2)\,du=2/3$. Thus
$$
\boxed{Q=\frac{\lambda a^5}{15}.}
$$
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