# Find the volume of the solid generated by rotating about the x-axis th

Find the volume of the solid generated by rotating about the x-axis the region bounded by

and the x-axis.
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Uersfeldte
Step 1
Find the volume of the solid generated by rotating about the x-axis the region bounded by $y={4}^{x}$ where $x=-3$ and $x=3$ and the x-axis.
by applying boundary value analysis we can find the volume of solid.
Step 2
$y={4}^{x}$
Apply log on both sides
$\mathrm{log}\left(y\right)=\mathrm{log}\left({4}^{x}\right)$
$\mathrm{log}\left(y\right)=x\mathrm{log}\left(4\right)$
use implicit differentiation w.r. to 'x'
$\frac{1}{y}{y}^{\prime }=\mathrm{log}\left(4\right)$
${y}^{\prime }=y\mathrm{log}\left(4\right)$
${y}^{\prime }={4}^{x}\mathrm{log}\left(4\right)$
Now we apply boundries
${\int }_{-3}^{3}{y}^{\prime }={\int }_{-3}^{3}{4}^{x}\mathrm{log}\left(4\right)$
$=\mathrm{log}\left(4\right)\left[{4}^{\left(-3\right)}-{4}^{\left(3\right)}\right]$
$=\mathrm{log}\left(4\right)\left[\frac{1}{64}-64\right]$
$=\mathrm{log}\left(4\right)\left[0.015-64\right]$
$=63.98\mathrm{log}\left(4\right)$