<s>[INST] Volume of the solid cut by a plane. 
I'd like to find the volume of the following solid.
The solid enclosed by the paraboloid $z=4-x^2-y^2$ and the plane $x+y+z=1$.

Actually original problem is the following (I made upper problem...)
The solid enclosed by the cylinder $x^2+y^2=1$ and the plane $x+y+z=1$ and $z=-5$
In this case, 
(the volume of the solid)=$$
\int_{-1}^1 \int_{-\sqrt{1-x^2}}^\sqrt{1-x^2} \int_{-5}^{1-x-y}dzdydx$$
Now, in case of

The solid enclosed by the paraboloid $z=4-x^2-y^2$ and the plane $x+y+z=1$.

How can I solve this?
Could you give me some hint, please?
 [/INST] $$\begin{align}
z & = 4 - x^2 - y^2\\
1 - x - y & = 4 - x^2 - y^2\\
0 & = 3 - x^2 + x - y^2 - y\\
0 & = 3 - x^2 + x - \frac14 + \frac14 - y^2 + y - \frac14 + \frac14\\
0 & = 3 - (x - \frac12)^2 + \frac14 - (y - \frac12)^2 + \frac14\\
\frac72 & = (x - \frac12)^2 + (y - \frac12)^2
\end{align}$$
So the region of intersection between the plane and paraboloid lies over a circle of radius $\sqrt{\frac72}$ centered at $(\frac12, \frac12)$.  Now you can set up the bounds of a 2-d integral.
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