Double Bubble

For more information about such sphere intersection and the derivation of given formulae, click here

If two spheres have radii $r_1$ and $r_2$ and if their centers are $d$ units apart, then the volume enclosed by them is $V=\frac{(r_1+r_2-d)^2(d^2+2d(r_1+r_2)-3(r_1-r_2)^2)}{12d}\pi$ In case of this question, $r_1=3$ , $r_2=5$ and $d=6$ .

So, the enclosed volume is $V=\frac{(3+5-6)^2(6^2+2(6)(3+5)-3(3-5)^2)}{12\cdot6}\pi=\frac{20}{3}\pi$ Hence, $a=20$ and $b=3$ and the value $\frac{a+1}{b}=\frac{20+1}{3}= \boxed{7}$