 # Do the following for the given curve on the given interval. a. Set up an integral for pancha3 2021-09-11 Answered
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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$\frac{dy}{dx}=\frac{d}{dx}\left(\mathrm{sin}\left\{x\right\}-\mathrm{cos}\left\{x\right\}\right)$
$=\mathrm{cos}\left\{x\right\}-\left[x\left(-\mathrm{sin}\left\{x\right\}\right)+\mathrm{cos}\left\{x\right\}\right]$
$=\mathrm{cos}\left\{x\right\}+2\mathrm{sin}\left\{x\right\}+\mathrm{cos}\left\{x\right\}$
$=2\mathrm{cos}\left\{x\right\}+x\mathrm{sin}\left\{x\right\}$
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) $\lambda ={\int }_{0}^{2\pi }\sqrt{1+{\left(\frac{dy}{dx}\right)}^{2}}$
$={\int }_{0}^{2\pi }\sqrt{1+{\left(2\mathrm{cos}\left\{x\right\}+x\mathrm{sin}\left\{x\right\}\right)}^{2}}dx$
$={\int }_{0}^{2\pi }\sqrt{1+4{\mathrm{cos}}^{2}\left\{x\right\}+{x}^{2}{\mathrm{sin}}^{2}\left\{x\right\}+4x\mathrm{cos}\left\{x\right\}\mathrm{sin}\left\{x\right\}}$
${\left[\frac{1}{2\sqrt{1+4{\mathrm{cos}}^{2}\left\{x\right\}+{x}^{2}<}}}_{}^{}$