Shell Method for $y=1/x$

Question

I understand shell, at least I thought I did. I tried doing $\int_0^.144(y)(1)+\int_.144^.168(y)(1/y)$ but the video showed the cylinder height to be $1-1/y$. Neither seem to be giving me decent numbers. Are the limits different?

Answer 1

First avoid decimals. You are sure to get rounding errors and after a while they add up to enough to make your result undependable. Do all the work with fractions as long as they are one or two digits, then if necessary change to a decimal at the end.

I see the first part (large yellow sample slice) is a cylinder (on it side) height = 6-7 = 1 and radius = 1/7. Volume by simple geometry = $\pi r^2 h = \pi \cdot 1/49 \cdot 1 = \pi/49$.

By shells that volume is $ \int_{y=0}^{1/7} 2 \pi \cdot y \cdot 1 dy = 2\pi y^2 / 2 ]_{y=0}^{1/7}$ which gives the same result $\pi /49$.

The second part has each shell radius = $2 \pi y \ $ again, from $y = 1/7 $ to $y = 1/6$, thickness $dy$, and "height" of shell (horizontal here, see small yellow sample stripe in the upper part of the diagram) = $x - 6$. Since $y = 1/x$ then $x = 1/y$ so the "height" of the shell is $(x - 6) =(1/y - 6)$.

It helps to draw these out. Unfortunately I have not yet found how to uload sketches here so please try t draw the shell yourself and label it to clarify.

Volume of this second part, by shells = $\int_{y=1/7}^{1/6} 2 \pi y (1/y - 6) dy$

$=\int_{y=1/7}^{1/6} (2 \pi - 12 \pi y) dy = (2 \pi y - 12 \pi y^2 / 2) \ ]_{y=1/7}^{1/6} = 2\pi (1/6 - 1/7) - 6\pi ((1/6)^2 - (1/7)^2$

=$2 \pi ( 1/42 - 6 \pi (49-36)/(36 \cdot 49) = \pi /21 - 13 \pi / 294$

Adding the first part above, total volume = $\pi/49 + \pi /21 - 13 \pi / 294$

Decimal approximation $V \approx .07480$

(Always double-check all calculations; good but nobody's perfect)