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,