Calculus: Early Transcendentals, Chapter 1, 1.1, Section 1.1, Problem 26

The amount of air required to inflate a spherical balloon from radius r to radius (r+1) is required:
The difference in the volumes is:
V_2-V_1=4/3 pi (r+1)^3-4/3 pi r^3
=4/3 pi[r^3+3r^2+3r+1-r^3]
=4/3 pi[3r^2+3r+1]
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Given an initial radius r, the amount of air A needed to increase the radius of a spherical balloon is:
A=4/3 pi(3r^2+3r+1)
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