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Hi! I am currently working on some calc2 online homework problems and I am having difficulty with this particular question. To be completely honest I am not sure how to even approach this problem, so if someone would be kind enough to help me solve this one I would really appreciate it, Thank you!

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  • $\begingroup$ There are two loops in the diagram. Each loop is traced out once from one zero of $25 \cos 2\theta$ to the next. What are two consecutive zeroes of $25 \cos 2\theta$? If you have these limits of integration, can you finish the integral in polar coordinates? $\endgroup$ – Eric Towers Apr 28 '14 at 0:11
  • $\begingroup$ I am still not sure how to continue to solve this problem. I am really lost in my calc class. Do you think you could possibly help me solve this one step by step so that I can understand it better? $\endgroup$ – user124539 Apr 28 '14 at 0:18
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The lemniscate will pass through the origin when $r = 0$, i.e., when $\cos(2\theta) = 0$.This will happen at infinitely many values of $\theta$, but to we want to find two values of $\theta$ between which one loop is traced. Thus we can choose the values of $\theta$: $\theta = \frac{-\pi}{4}$ and $\theta = \frac{\pi}{4}$.

To get the area we then integrate the following:

$A = \displaystyle\int_{\frac{-\pi}{4}}^{\frac{\pi}{4}} \frac{25}{2}\cos(2\theta) d\theta$

$A = 12.5$

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