Find the increment Δ V of the volume of a sphere when

Find the increment $\Delta V$ of the volume of a sphere when the radius changes by $\Delta r$. Compute $\Delta V$ and $dV$ for $r = 3$ with 1° $\Delta r = 0.5$; 2° $\Delta r = 0.1$; 3° $\Delta r = 0.01$, and determine the error made in replacing the increment by the differential.

Step 1: The exact increment

With $V = \dfrac{4}{3}\pi r^{3}$ and $\bigl(r+\Delta r\bigr)^{3} = r^{3}+3r^{2}\Delta r+3r(\Delta r)^{2}+(\Delta r)^{3}$: \[\Delta V = \frac{4\pi}{3}\Bigl[3r^{2}\Delta r+3r(\Delta r)^{2}+(\Delta r)^{3}\Bigr] = 4\pi r^{2}\Delta r+4\pi r(\Delta r)^{2}+\frac{4\pi}{3}(\Delta r)^{3}.\]

Step 2: The differential

\[ dV = V'_{r}\,\Delta r = 4\pi r^{2}\,\Delta r. \]

This is the surface area times the thickness — the volume of a thin spherical shell, which is exactly what the linear term should be.

Step 3: The error in closed form

\[ \bigl|\Delta V-dV\bigr| = 4\pi r(\Delta r)^{2}+\frac{4\pi}{3}(\Delta r)^{3} = \frac{4\pi(\Delta r)^{2}\bigl(3r+\Delta r\bigr)}{3}. \]

At $r = 3$ the formulas become

\[ \Delta V = 36\pi\,\Delta r+12\pi(\Delta r)^{2}+\frac{4\pi}{3}(\Delta r)^{3}, \qquad dV = 36\pi\,\Delta r, \qquad \bigl|\Delta V-dV\bigr| = \frac{4\pi(\Delta r)^{2}\bigl(9+\Delta r\bigr)}{3}. \]

Step 4: Case 1°: $\Delta r = 0.5$

\[ \Delta V = 18\pi+3\pi+\frac{\pi}{6} = \frac{127\pi}{6} \approx 66.497, \qquad dV = 18\pi \approx 56.549, \]

\[ \bigl|\Delta V-dV\bigr| = \frac{19\pi}{6} \approx 9.948, \qquad \frac{19}{127} \approx 14.96\%. \]

Step 5: Case 2°: $\Delta r = 0.1$

\[ \Delta V = 3.6\pi+0.12\pi+0.0013\overline{3}\pi \approx 11.6909, \qquad dV = 3.6\pi \approx 11.3097, \]

\[ \bigl|\Delta V-dV\bigr| \approx 0.3812, \qquad \frac{0.3812}{11.6909} \approx 3.26\%. \]

Step 6: Case 3°: $\Delta r = 0.01$

\[ \Delta V \approx 1.13475, \qquad dV = 0.36\pi \approx 1.13097, \]

\[ \bigl|\Delta V-dV\bigr| = \frac{4\pi(0.01)^{2}(9.01)}{3} \approx 0.003774, \qquad \frac{0.003774}{1.13475} \approx 0.33\%. \]

Step 7: Summary

\[\begin{array}{c|c|c|c|c} \Delta r & \Delta V & dV & \text{absolute} & \text{relative} \\ \hline 0.5 & 66.497 & 56.549 & 9.948 & 14.96\% \\ 0.1 & 11.6909 & 11.3097 & 0.3812 & 3.26\% \\ 0.01 & 1.13475 & 1.13097 & 0.003774 & 0.33\% \end{array}\]

\[\text{Direct check, } \Delta r = 0.5:\ \frac{4\pi}{3}\bigl(3.5^{3}-3^{3}\bigr) = \frac{4\pi}{3}\bigl(42.875-27\bigr) = \frac{4\pi\cdot 15.875}{3} \approx 66.497 \;\;\checkmark\]

\[\text{Relative error } \approx \frac{\Delta r}{r} \text{ for small } \Delta r:\ \frac{0.01}{3} = 0.33\% \;\;\checkmark\]