Answer
$\frac{4}{3}\pi (r+3)^3-\frac{4}{3}\pi r^3$
Work Step by Step
Step 1. Given $V(r)=\frac{4}{3}\pi r^3$, for $r\to r+3$, we have $V(r+3)=\frac{4}{3}\pi (r+3)^3$
Step 2. The volume gain is $V'(r)=\Delta V=V(r+3)-V(r)=\frac{4}{3}\pi (r+3)^3-\frac{4}{3}\pi r^3$