Answer to Question #176667 in Calculus for Joshua

Question

Use the method of cylinders to determine the volume of the solid obtained by rotating the region bounded by y=1/x, x=1/2, x=4 and the x-axis about the y-axis.

Answer 1

The volume of the solid generated by rotating this region about the y- axis may be found by adding the volumes of the solids generated by rotating the region bounded by $x=4$, $x= \frac 12$, $y=\frac1x$ (v₁) and the region bounded by $x=4$, $x=\frac 12$, $y=1/4$ and the x-axis (v₂).

$v_1=2π∫_0^{\frac14}yx dy$

$=2π∫_0^{\frac14}y(4-\frac 12) dy$

$=2π∫_0^{\frac14)} {\frac72} y dy$

$= [ \frac {7π y^2}2]^{\frac14}_0$

$=7π [\frac{(\frac14)^2}2 -0]$

$=\frac 7{32}π cubic units of length$

$v_2=2π ∫_{\frac14}^2yx dy=2π∫_{\frac14}^2y(\frac1y- \frac 12)dy$

$=2π∫_{\frac14}^2(1- \frac12 y)dy$

$=2π[y-\frac14 y^2 ] ^2_{\frac14}$

$=2π[(2-\frac 14 (2)^2 )-(\frac 14-\frac 14 (\frac14)^2)]$

$=2π(1- \frac {15}{64})$

$=\frac {49}{32} π cubic units of length$

$The total volume V=v_1+v_2$

$=\frac7{32}π+ \frac {49}{32} π$

$=\frac 74 π cubic units of length$