Single Variable Calculus, Chapter 1, 1.1, Section 1.1, Problem 22

The volume of a spherical balloon with radius r inches is equal to $V(r)= \frac{4}{3} \pi r^3$. Find a function that represents the amount of air required to inflate the balloon from a radius of r inches to radius of $r+1$ inches.



The amount of air required to inflate the balloon is equal to the volume of the sphere with new radius minus the volume of the balloon with the original radius.



$\begin{equation} \begin{aligned} V_{Total} &= V_{(r+1)} - V_{(r)}\\ \\ \displaystyle V_{Total} &= \frac{4}{3}\pi \, (r+1)^3 - \frac{4}{3} \pi r^3 && (\text{Expanding the cubic function})\\ \\ \displaystyle V_{Total}&= \frac{4}{3} \pi \left[ \cancel{r^3} + 3r^2 + 3r + 1 - \cancel{r^3} \right] && (\text{Simplifying the equation and combining like terms})\\ \end{aligned} \end{equation}$




$\fbox{$V_{Total}=\frac{4}{3} \pi \left[ 3r^2+3r+1 \right]$}$