# Determine the common region of these three sets in R^3:

I'm struggling a bit with finding the common region of these three regions (with the motivation that I will integrate its volume):

1) $R_1 = \{(x,y,z): x^2 + y^2 + z^2 \le 1\}$

2) $R_2 = \{(x,y,z): x^2 + y^2 \le a^2\}, 0<a<1$

3) $R_3 = \{(x,y,z): z\ge b(x^2 + y^2) \}, b>0$

for the integration, the problem also asks, "distinguish two cases:"

$ba^2\le \sqrt{1-a^2}$

$ba^2 > \sqrt{1-a^2}$

Any ideas are welcome.

I am able to isolate $x^2 + y^2$ in the inequalities that describe all three regions, but I don't know how to proceed from this point on. I am now merely just tripping up on getting bounds for $z$, and making other unproductive moves.

Thanks,

• if $b$ is small then the paraboloid hits the cylinder before it hits the sphere, which means the radius bound is $a$ and bottom and top are paraboloid and sphere. If $b$ is large then the paraboloid hits the sphere first, you need to calculate the radius bound, and the cylinder is not involved in any way. – Will Jagy Jan 7 '16 at 0:45
• Meanwhile, the cure for inability to visualize these things is drawing, in this case the $xz$ plane with $y = 0,$ printablepaper.net/category/graph – Will Jagy Jan 7 '16 at 0:48