Volume of a paraboloid and the plane

Question

Can someone help me start is problem off. Find the volume bounded by the paraboloid $x^2 + y^2 + z =7$ and the plane $z=1$.

Answer 1

Hint:

First, notice that the two surfaces intersect in the circle $$x^2+y^2=6,\quad z=1$$

Now, the region is symmetric respect to the planes $xz$ and $yz$, then it will be sufficient to determine the volume laying in the first octant: \begin{align} \frac{1}{4}V&=\int_0^{\sqrt 6}\int_0^{\sqrt{6-x^2}}\int_1^{7-x^2-y^2}\,dz\,dy\,dx \end{align} Where $V$ denotes the volume of the region.