## Calculus 8th Edition

$f(x)=\frac{4}{3}\pi (3r^{2}+3r+1)$
Lets calculate the amount of air needed to inflate a balloon with radius (r+1) and subtract the air needed for a balloon with radius r. $f(x)=V(r+1)-V(r)=\frac{4}{3}\pi (r+1)^{3}-\frac{4}{3}\pi r^{3}=\frac{4}{3}\pi ((r+1)^{3}-r^{3})$ *$(r+1)^{3}-r^{3}= r^{3}+3r^{2}+3r+1-r^{3}=3r^{2}+3r+1$ So, $f(x)=V(r+1)-V(r)=\frac{4}{3}\pi (r+1)^{3}-\frac{4}{3}\pi r^{3}=\frac{4}{3}\pi ((r+1)^{3}-r^{3})=\frac{4}{3}\pi (3r^{2}+3r+1)$