# Do the following for the given curve on the given interval. a. Set up an integral for

Do the following for the given curve on the given interval.
a. Set up an integral for the length of the curve.
b. Graph the curve to see what it looks like.
c. Use your​ graphers
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Macsen Nixon

$\frac{dy}{dx}=\frac{d}{dx}\left(\mathrm{sin}\left\{x\right\}-\mathrm{cos}\left\{x\right\}\right)$
$=\mathrm{cos}\left\{x\right\}-\left[x\left(-\mathrm{sin}\left\{x\right\}\right)+\mathrm{cos}\left\{x\right\}\right]$
$=\mathrm{cos}\left\{x\right\}+2\mathrm{sin}\left\{x\right\}+\mathrm{cos}\left\{x\right\}$
$=2\mathrm{cos}\left\{x\right\}+x\mathrm{sin}\left\{x\right\}$
a) set up integral
$L={\int }_{0}^{2\pi }{\sqrt{\left(1+\frac{dx}{dy}\right\}}}^{2}dx$
$={\int }_{0}^{2\pi }\sqrt{{\left(1+2\mathrm{cos}\left\{x\right\}+\left\{x\right\}\mathrm{sin}\left\{x\right\}\right)}^{2}}dx$
$={\int }_{0}^{2\pi }\sqrt{1+4\mathrm{cos}\left\{x\right\}+{x}^{2}{\mathrm{sin}}^{2}x+4x\mathrm{cos}\left\{x\right\}\mathrm{sin}\left\{x\right\}}$
b)

c) $\lambda ={\int }_{0}^{2\pi }\sqrt{1+{\left(\frac{dy}{dx}\right)}^{2}}$
$={\int }_{0}^{2\pi }\sqrt{1+{\left(2\mathrm{cos}\left\{x\right\}+x\mathrm{sin}\left\{x\right\}\right)}^{2}}dx$
$={\int }_{0}^{2\pi }\sqrt{1+4{\mathrm{cos}}^{2}\left\{x\right\}+{x}^{2}{\mathrm{sin}}^{2}\left\{x\right\}+4x\mathrm{cos}\left\{x\right\}\mathrm{sin}\left\{x\right\}}$
${\left[\frac{1}{2\sqrt{1+4{\mathrm{cos}}^{2}\left\{x\right\}+{x}^{2}<}}}_{}^{}$