Problems in understanding multiple integrals theory. I'm studying Double and Triple Integrals and stucked understanding how to determinate which reduction formula is ok for my case. Also got problems in finding $\rho$ and $\theta$ while switching in polar coordinates. Tried to solve this : 


*

*Calculate the volume |A|, with A = {$x$ $\in$ $\mathfrak R^3$ : $z$ $\ge$ $x^2+$ $y^2$, $z \le 1$, $x\ge z$}   


By setting $y = 0$, I drew in $z$ and $x$ axes functions to know which region to look and : 
Region is under green line $z=x$ and above $z=x^2$ in blue
Then :
$$\iiint_A dxdydz = \iint_Ddxdy\int_{x^2+y^2}^xdz  = \iint_D(x-x^2-y^2)dxdy$$
To determinate $D$ i'm setting     $x = x^2 + y^2$ who, by manipulation, is $(x-\frac{1}{2})^2 +$ $y^2$ = $\frac{1}{4}$ 
So a circle with center = $(\frac{1}{2},0)$ and radius = $\frac{1}{2}$  
Now by setting polar coordinates I end up with $0 \le\rho\le cos\theta$ and $0\le\theta\le\frac{\pi}{4}$. 
My final answer to this is $\frac{8+3\pi}{384}$. Result is different and it's $\frac{3\pi}{128}$. 
I need a comparison just to know if I'm doing this thing in a right way.
Thank you for all your assistance.
 A: $\newcommand{\bbx}[1]{\,\bbox[8px,border:1px groove navy]{\displaystyle{#1}}\,}
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Here after, $\ds{\bracks{\cdots}}$ is an
  Iverson Bracket.  


As you can see, the region '$x > 1$' has empty intersection with '$x^{2} + y^{2} < z < 1$'. So, your integration is solely over a region with '$x < 1$' and '$x^{2} + y^{2} < z < x$'. 
\begin{align}
\iiint_{A}\dd x\,\dd y\,\dd z & =
\iiint_{\large\mathbb{R}^{3}}\bracks{x^{2} + y^{2} \leq z \leq \min\braces{1,x}}
\dd x\,\dd y\,\dd z
\\[1cm] & =
\iiint_{\large\mathbb{R}^{3}}\bracks{x \leq 1}
\bracks{x^{2} + y^{2} \leq z \leq x}\dd x\,\dd y\,\dd z
\\[5mm] & +
\iiint_{\large\mathbb{R}^{3}}\bracks{x \geq 1}
\bracks{x^{2} + y^{2} \leq z \leq 1}\dd x\,\dd y\,\dd z
\\[1cm] & =
\int_{-\infty}^{\infty}\int_{-\infty}^{1}\bracks{x^{2}  + y^{2} \leq x}
\int_{x^{2} + y^{2}}^{x}\dd z\,\dd x\,\dd y
\\ & +
\overbrace{\int_{-\infty}^{\infty}\int_{1}^{\infty}
\bracks{x^{2}  + y^{2} \leq 1}
\int_{x^{2} + y^{2}}^{1}\dd z\,\dd x\,\dd y}^{\ds{=\ 0}}
\\[1cm] & =
\int_{-\infty}^{\infty}\int_{-\infty}^{1/2}
\bracks{x^{2}  + y^{2} \leq {1 \over 4}}
\int_{x^{2} + x + 1/4 + y^{2}}^{x + 1/2}\dd z\,\dd x\,\dd y =
\int_{0}^{2\pi}\int_{0}^{1/2}\pars{{1 \over 4} - r^{2}}r\,\dd r\,\dd\phi
\\[5mm] & =
\bbx{\ds{\pi \over 32}}
\end{align}
