MATH 152 Chapter 7.3

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Volume by Cylindrical Shells

Volume of a shell = circumference × height × thickness When unrolled into a rectangular prism:

V=lwh

$l=2\pi r$ r is y or x
$w=\Delta r_{i}=\mathrm {d} r$
$h=h$ x or y involved: g(y) or f(x)

V=2\pi rh\,\mathrm {d} r

or

V=2\pi rf(r)\,\mathrm {d} r

if the radius is horizontal, then

r=x-(axis)

if axis is at

x=-1

, then

r=x-(-1)=x+1

\mathrm {d} r=\mathrm {d} x

h=y_{2}-y_{1}

(top — bottom)

NOTE: boundaries are for the radius: the range is from the center to the edge of the region.

if the radius is vertical, then

r=y-(axis)

\mathrm {d} r=\mathrm {d} y

h=x_{2}-x_{1}

(right — left)

Monday, September 13, 2010

Slices integrate with respect to height variable. Shells integrate with respect to the thickness variable (same direction as radius).

	Rotation about a horizontal axis ( $x$ )	Rotation about a vertical axis ( $y$ )
$y=f(x)$	Slices $V=\pi \int {y^{2}\,\mathrm {d} x}$	Shells $V=2\pi \int {xy\,\mathrm {d} x}$
$x=g(y)$	Shells $V=2\pi \int {yx\,\mathrm {d} y}$	Slices $V=\pi \int {x^{2}\,\mathrm {d} y}$