The Shell Method

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We practice setting up setting up volume calculations using the shell method.

(Video) Calculus: Single Variable

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The region defined by the inequalities $-\sqrt {1-x^2} \leq y \leq \sqrt {1-x^2}$ and $x \geq 0$ (shown below) is revolved around the $y$ -ais. Compute the volume using the shell method.

When the slicing variable is $x$ , the radius of a shell is the
distance from an $x$ -slice to the axis $x = 0$ . Thus $r(x) = \answer {x} - \answer {0}.$
The height of an $x$ -slice is equal to
The volume is equal to the integral of $2 \pi r h$ , so $V = \int _{\answer {0}}^{\answer {1}} \answer {4 \pi x \sqrt {1-x^2}} dx = \answer {\frac {4\pi }{3}}.$ (Note: to compute the integral, we can make the substitution $u = 1-x^2$ .)

The region between the curves $x = \sqrt {y}$ and $x = y + \sqrt {y}$ from $y=0$ to $y=1$ is revolved around the axis $y=1$ . Compute the volume of the resulting solid.

When the slicing variable is $y$ , the radius of a shell is the
distance from a $y$ -slice to the axis $y = 1$ . Thus $r(y) = \answer {1} - \answer {y}.$
The “height” of a $y$ -slice is equal to
The volume is equal to the integral of $2 \pi r h$ , so $V = \int _{\answer {0}}^{\answer {1}} \answer {2 \pi y(1-y)} dy = \answer {\frac {\pi }{3}}.$

Press...	...to do
left/right arrows	Move cursor
shift+left/right arrows	Select region
ctrl+a	Select all
ctrl+x/c/v	Cut/copy/paste
ctrl+z/y	Undo/redo
ctrl+left/right	Add entry to list or column to matrix
shift+ctrl+left/right	Add copy of current entry/column to to list/matrix
ctrl+up/down	Add row to matrix
shift+ctrl+up/down	Add copy of current row to matrix
ctrl+backspace	Delete current entry in list or column in matrix
ctrl+shift+backspace	Delete current row in matrix

Type...	...to get
norm	$\|\|\blue{[?]}\|\|$
text	$\text{\blue{[?]}}$
sym_name	$\backslash\texttt{\blue{[?]}}$
abs	$\left\|\blue{[?]}\right\|$
sqrt	$\sqrt{\blue{[?]}}$
paren	$\left(\blue{[?]}\right)$
floor	$\lfloor \blue{[?]} \rfloor$
factorial	$\blue{[?]}!$
exp	${\blue{[?]}}^{\blue{[?]}}$
sub	${\blue{[?]}}_{\blue{[?]}}$
frac	$\dfrac{\blue{[?]}}{\blue{[?]}}$
int	$\displaystyle\int{\blue{[?]}}d\blue{[?]}$
defi	$\displaystyle\int_{\blue{[?]}}^{\blue{[?]}}\blue{[?]}d\blue{[?]}$
deriv	$\displaystyle\frac{d}{d\blue{[?]}}\blue{[?]}$
sum	$\displaystyle\sum_{\blue{[?]}}^{\blue{[?]}}\blue{[?]}$
prod	$\displaystyle\prod_{\blue{[?]}}^{\blue{[?]}}\blue{[?]}$
root	$\sqrt[\blue{[?]}]{\blue{[?]}}$
vec	$\left\langle \blue{[?]} \right\rangle$
mat	$\left(\begin{matrix} \blue{[?]} \end{matrix}\right)$
*	$\cdot$
infinity	$\infty$
arcsin	$\arcsin\left(\blue{[?]}\right)$
arccos	$\arccos\left(\blue{[?]}\right)$
arctan	$\arctan\left(\blue{[?]}\right)$
sin	$\sin\left(\blue{[?]}\right)$
cos	$\cos\left(\blue{[?]}\right)$
tan	$\tan\left(\blue{[?]}\right)$
sec	$\sec\left(\blue{[?]}\right)$
csc	$\csc\left(\blue{[?]}\right)$
cot	$\cot\left(\blue{[?]}\right)$
log	$\log\left(\blue{[?]}\right)$
ln	$\ln\left(\blue{[?]}\right)$
alpha	$\alpha$
beta	$\beta$
gamma	$\gamma$
delta	$\delta$
epsilon	$\epsilon$
zeta	$\zeta$
eta	$\eta$
theta	$\theta$
iota	$\iota$
kappa	$\kappa$
lambda	$\lambda$
mu	$\mu$
nu	$\nu$
xi	$\xi$
omicron	$\omicron$
pi	$\pi$
rho	$\rho$
sigma	$\sigma$
tau	$\tau$
upsilon	$\upsilon$
phi	$\phi$
chi	$\chi$
psi	$\psi$
omega	$\omega$
Gamma	$\Gamma$
Delta	$\Delta$
Theta	$\Theta$
Lambda	$\Lambda$
Xi	$\Xi$
Pi	$\Pi$
Sigma	$\Sigma$
Phi	$\Phi$
Psi	$\Psi$
Omega	$\Omega$

(Video) Calculus: Single Variable

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