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. 
 A: 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 $\frac{\sqrt{6}}{2}$.
A: These are algebraic equations/inequations of the second degree in 2D and 3D, better known as conics and quadrics. You should get some familiarity with them and learn to recognize the different types (ellipse, parabola, hyperbola, and cylinders, ellipsoids, paraboloids, hyperbolic paraboloid, hyperboloids) http://en.wikipedia.org/wiki/Conic http://en.wikipedia.org/wiki/Quadric.
In the first example given, you will recognize a circle as having no $xy$ coefficient and equal $x^2$ and $y^2$ terms; the presence of first degree terms indicate it is not centered on the origin. In the second example, you will recognize an hyperboloid of revolution, from two second degree terms of the same coefficient and a third of opposite sign.
