How to plot region of integration in Mathematica?

Transform your conditions to cylindrical coordinates

cond = 
  x^2 + (y - 1)^2 < 1 &&0 < z < x^2 + y^2 /. {x -> r Cos[φ], y -> r Sin[φ]} // 
    FullSimplify[#, {r > 0, -Pi < φ < Pi}] &

(*r < 2 Sin[φ] && 0 < z < r^2*)

to get the integration limits!

The first condition (remember r > 0) implies 0 < φ < Pi.

The integration limits follow to

{φ, 0, Pi}, {r, 0, 2 Sin[φ]}, {z, 0, r^2}

Checking the results:

Volume of the cartesian region:

ImplicitRegion[x^2 + (y - 1)^2 < 1 && 0 < z < x^2 + y^2, {x, y, z}] // Volume

(*3Pi/2*)

equals

Integrate[r, {φ, 0, Pi}, {r, 0, 2 Sin[φ]}, {z, 0,r^2}]

(* 3Pi/2*)

That's it. Hope it helps.

You can visualize the region of integration as follows if specified in rectangular / Cartesian coordinates. I am looking for a way to specify in cylindrical to the plot directly and will update when I find it.

With[{
    Δ=0.1
},
    RegionPlot3D[And[
        x^2+(y-1)^2<=1,
        z<=x^2+y^2,
        z>=0
    ],
        {x,-1-Δ,1+Δ},
        {y,0-Δ,2+Δ},
        {z,0-Δ,4+Δ},

        Mesh->10,
        MeshFunctions->{#3&},
        PlotStyle->Directive[Opacity[0.5],Yellow],
        MeshShading->{Red,Automatic},

        PlotPoints->150,

        PlotTheme->"Detailed",
        AxesLabel->Automatic
    ]
]

---

ClearAll[getCartesian,getCylindrical];
getCartesian[field_]:=FullSimplify@TransformedField["Cylindrical"->"Cartesian",field,{r,θ,\[ScriptZ]}->{x,y,z}];
getCylindrical[field_]:=FullSimplify@TransformedField["Cartesian"->"Cylindrical",field,{x,y,z}->{r,θ,\[ScriptZ]}];

getCylindrical/@And[
    x^2+(y-1)^2<=1,
    z<=x^2+y^2,
    z>=0
]

r^2 <= 2 r Sin[θ] && [ScriptZ] <= r^2 && [ScriptZ] >= 0

Now you can use this transformation function to directly specify the conditions in cylindrical coordinates and it will be plotted.

With[{
    Δ=0.1
},
    RegionPlot3D[Evaluate[getCartesian/@And[
        r^2<=2r Sin[θ],
        \[ScriptZ]<=r^2,
        \[ScriptZ]>=0
    ]],
        {x,-1-Δ,1+Δ},
        {y,0-Δ,2+Δ},
        {z,0-Δ,4+Δ},

        Mesh->10,
        MeshFunctions->{#3&},
        PlotStyle->Directive[Opacity[0.5],Yellow],
        MeshShading->{Red,Automatic},

        PlotPoints->150,

        PlotTheme->"Detailed",
        AxesLabel->Automatic
    ]
]