Geometric intuition: Seeing the regions in double integrals Context: solving double integrals.
I had the formula $$x^2+y^2=1-x-y$$ yet I could not see what shape it had. This is even more true with 3D pictures like $$2x^2+2y^2 \le 1+z^2.$$ Is there a summary somewhere of shapes to learn, so that I can get this. 

Note 
\begin{align}
x^2+y^2&=1-x-y \\
\Leftrightarrow x^2+y^2+x+y & =1 \\
\Leftrightarrow x^2+x+\frac{1}{4}+y^2+y+\frac{1}{4}&=1+\frac{1}{4}+\frac{1}{4} \\
\Leftrightarrow \left(x+\frac{1}{2}\right)^2+ \left(y+\frac{1}{2}\right)^2 & =\frac{3}{2}
\end{align}
Last equation describes a circular cilinder with axis parallel to $z$ axis and radius $\frac{\sqrt{6}}{2}$.
