First of all we see from $x^2+y^2=9$ that $y$ lives in $J_y=[-3,3]$.
Now we can compute using Fubini:
$$
\begin{aligned}
&\iiint_D\sqrt{10+y^2-x^2}\;dx\;dy\;dz
\\
&\qquad=
\int_{y\in[-3,3]}dy
\int_{x\in J_x(y)}dx
\int_{z\in J_z(x,y)}\sqrt{10+y^2-x^2}\;dz
\\
&\qquad=
\int_{y\in[-3,3]}dy
\int_{x\in J_x(y)}dx
\cdot \left(\text{length of $J_z(x,y)$}\right)\cdot\sqrt{10+y^2-x^2}
\\
&\qquad=
\int_{y\in[-3,3]}dy
\int_{x\in J_x(y)}dx
\cdot 2\sqrt{10+y^2-x^2}\cdot\sqrt{10+y^2-x^2}
\\
&\qquad=
2\int_{y\in[-3,3]}dy
\int_{x\in J_x(y)}dx
\cdot (10+y^2-x^2)
\\
&\qquad=
2\int_{y\in[-3,3]}dy
\cdot
\left[\ 10x + xy^2-\frac 13x^3\ \right]
_{x=-\sqrt{9-y^2}}
^{x=+\sqrt{9-y^2}}
\\
&\qquad=
4
\int_{y\in[-3,3]}dy
\cdot
\sqrt{9-y^2}\cdot
\left( \ 10 + y^2-\frac 13(9-y^2)\ \right)
\\
&\qquad=
4
\int_{-3}^3
\sqrt{9-y^2}\cdot
\left( \ 7 + \frac 43y^2\ \right)\; dy
\\
&\qquad\qquad\text{Substitution: } y=3\sin t\ ,
\\
&\qquad=
4
\int_{-\pi/2}^{\pi/2}
3\cos t\cdot
\left( \ 7 + \frac 43\cdot 9\sin^2 t\ \right)\; 3\cos t\; dt
\\
&\qquad=
4\cdot 3^2\cdot 2
\int_0^{\pi/2}
\cos^2 t\cdot
\left( \ 7 + 12\sin^2 t\ \right)\; dt
\\
&\qquad=180\pi\ .
\end{aligned}
$$
Computer check, here sage:
sage: integral(
....: sqrt(10+y^2-x^2), z, -sqrt(10+y^2-x^2), +sqrt(10+y^2-x^2) )
-2*x^2 + 2*y^2 + 20
sage: integral( integral(
....: sqrt(10+y^2-x^2), z, -sqrt(10+y^2-x^2), +sqrt(10+y^2-x^2) ),
....: x, -sqrt(9-y^2), +sqrt(9-y^2) )
4/3*(4*y^2 + 21)*sqrt(-y^2 + 9)
sage: integral( integral( integral(
....: sqrt(10+y^2-x^2), z, -sqrt(10+y^2-x^2), +sqrt(10+y^2-x^2) ),
....: x, -sqrt(9-y^2), +sqrt(9-y^2) ),
....: y, -3, 3 )
180*pi
Later edit:
Using cylindrical coordinates is possible, and this leads to a similar computation. I was avoiding this first because using $x=r\cos t$, $y=r\sin t$ we do not have an immediat simplification for $\sqrt{10+y^2-x^2}\ge 1$.
Note however that the correct condition for $z$, after we fix $x,y$ and/or after we fix the corresponding values for $r,t$, is
$$
\begin{aligned}
|z| &\le \sqrt{10+y^2-x^2}\ ,\text{ respectively}\\
|z| &\le \sqrt{10+r^2(\sin^2 t-\cos^2 t)}
=\sqrt{10-r^2\cos2t}
\ ,
\end{aligned}
$$
so we would have to write
$$
\begin{aligned}
&\iiint_D\sqrt{10+y^2-x^2}\;dx\;dy\;dz
\\
&\qquad=
\int_{(x,y)\in\text{Disk of radius $3$ centered in origin}}dx\;dy
\int_{z\in J_z(x,y)}\sqrt{10+y^2-x^2}\;dz
\\
&\qquad=
\int_{(r,t)\in[-3,3]\times[0,2\pi]}r\;dr\;dt
\int_{z\in [-\sqrt{10-r^2\cos 2t},\ +\sqrt{10-r^2\cos 2t}]}
\sqrt{10-r^2\cos 2t}\;dz
\\
&\qquad=
\int_{(r,t)\in[-3,3]\times[0,2\pi]}r\;dr\;dt
\cdot(\text{ length of $[-\sqrt{10-r^2\cos 2t},\ +\sqrt{10-r^2\cos 2t}]$ })\cdot
\sqrt{10-r^2\cos 2t}
\\
&\qquad=
\int_{(r,t)\in[-3,3]\times[0,2\pi]}r\;dr\;dt
\cdot2(10-r^2\cos 2t)
\\
&\qquad=
\int_{(r,t)\in[-3,3]\times[0,2\pi]}r\;dr\;dt
\cdot2\cdot 10
\\
&\qquad\qquad\text{ since the integral of $\cos 2t$ vanishes on $[0,2\pi]$,}
\\
&\qquad=
\int_0^32\pi\cdot r\;dr
\cdot2\cdot 10
\\
&\qquad=
2\pi\cdot \frac 92 \cdot2\cdot 10
\\
&\qquad=
180\pi\ .
\end{aligned}
$$
If you want use a triple integral than note that, since we don't have a cylindrical symmetry, the use of cylindrical coordinates is not the best choice.
It is simpler to use Cartesian coordinates, noting that:
$$
-\sqrt{1-(y-2)^2}\le x \le \sqrt{1-(y-2)^2}
$$
for: $1\le y\le3$
so the integral is:
$$
\int_1^3\int_{-\sqrt{1-(y-2)^2}}^{\sqrt{1-(y-2)^2}}\int_0^2 dzdxdy
$$
Best Answer
I would recommend using rectangular/Cartesian coordinates here. Cylindrical coordinates are going to be a bit of a hassle when we get to working with finding the bounds for $r$ and $\theta$ on the $yz$-plane.
So for this problem, we know that since we're in the first octant, we know that each of $x,y,z$ are nonnegative. You have the right idea of establishing bounds for $z$ here, but remember, the equation that you're given is simply $y^2+z^2=4$, not $x^2+y^2+z^2=4$, so you'll want to use $\sqrt{4-y^2}$ as your upper bound. (so that $0\le z\le \sqrt{4-y^2}$)
For your other two bounds, you'll want to project your given planes as well as your bounds for $z$ onto the $xy$-plane (where $z=0$). From there, since you'll want to have that $\sqrt{4-y^2}$ is nonnegative (or, if you prefer, $y$ must satisfy the equation $y^2+z^2=4$), we have that $-2\le y\le 2$, and since we're working in the first octant, this means we can further restrict our bounds for $y$ to $0\le y\le 2$. With this and your bounds for $x$ as given by the problem statement (so that $0\le x\le y$), we have that we can obtain the following integral: $$\int^2_0\int^y_0\int^{\sqrt{4-y^2}}_0z\text{ }dz\text{ }dx\text{ }dy$$ which, after going through the Calculus, should evaluate to 2.