Arc Length

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We practice setting up and executing arc length calculations.

(Video) Calculus: Single Variable

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Compute the arc length of the curve

$x = -\frac {1}{8}y^{-2} - \frac {1}{4}y^{4} - 2$ between the endpoints $y = 1/\sqrt {2}$ and $y = 1$ .

First we compute the derivative: $\frac {dx}{dy} = \answer {\frac {1}{4} y^{-3} - y^3}.$
Next, we write the arc length element: $ds = \sqrt { 1 + \left (\frac {dx}{dy}\right )^2} dy = \sqrt {1 + \left ( \answer { \frac {1}{4} y^{-3} - y^3} \right )^2} dy.$
The key to integrating is to first fully simplify the quantity inside the square root: $\begin {aligned} 1 + \left ( \answer {\frac {1}{4} y^{-3} - y^3} \right )^2 = & 1 + \frac {1}{16} y^{-6} + 2 \left ( \frac {1}{4} y^{-3} \right ) \left ( - y^3 \right ) + y^6 \\ & = 1 + \frac {1}{16} y^{-6} - \frac {1}{2} + y^6 \end {aligned}$
When the middle term in a binomial (i.e., FOIL) expansion is $-1/2$ adding one always gives a perfect square. In this case, the algebraic fact is $1 + \frac {1}{16} y^{-6} - \frac {1}{2} + y^6 = \frac {1}{16} y^{-6} + \frac {1}{2} + y^6 = \left ( \answer {\frac {1}{4} y^{-3} + y^3} \right )^2.$ The general phenomenon is that what was formerly the square of a difference—in this case, $\left ( \frac {1}{4} y^{-3} - y^3 \right )^2$ —becomes the square of the corresponding sum.
Therefore $\begin {aligned} L & = \int _{1/\sqrt {2}}^{1} \sqrt {1 + \left ( \frac {dx}{dy} \right )^2}~dy = \int _{1/\sqrt {2}}^{1} \sqrt {1 + \left ( \answer {\frac {1}{4}y^{-3} - y^{3}} \right )^2}~dy \\ & = \int _{1/\sqrt {2}}^{1} \sqrt {1 + \frac {1}{16}y^{-6} - \frac {1}{2} + y^{6}} ~ dy = \int _{1/\sqrt {2}}^{1} \sqrt { \left ( \answer {\frac {1}{4}y^{-3} + y^{3}} \right )^2} ~ dy \\ & = \int _{1/\sqrt {2}}^{1} \answer {\frac {1}{4}y^{-3} + y^{3}} ~ dy = \left . \left ( \answer {-\frac {1}{8}y^{-2} + \frac {1}{4}y^{4}} \right ) \right |_{1/\sqrt {2}}^{1} \\ & = \left ( -\frac {1}{8} + \frac {1}{4} \right ) - \left ( -\frac {1}{4} + \frac {1}{16} \right ) = \frac {5}{16}. \end {aligned}$

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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	$\Omega$

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