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I have 4 curves, and I need to find the area between the curves.

  • $\ln x$
  • $\\|\ln x|$
  • $\ln |x|$
  • $\\|\ln |x||$

enter image description here

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    $\begingroup$ Good luck, then. And what was your question? $\endgroup$
    – user436658
    Feb 15 '18 at 7:27
  • $\begingroup$ I need to find the area enclosed $\endgroup$ Feb 15 '18 at 8:03
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$$\int_0^1\ln x\mathrm{d}x=x\ln x-x\bigg|_0^1=-1$$ $$Area=-4\int_0^1\ln x\mathrm{d}x=4$$

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  • $\begingroup$ I think he is asking for the area, not integral value. If he asks about integral value, I think the answer is 0. $\endgroup$
    – liaoyulei
    Feb 15 '18 at 7:41
  • $\begingroup$ @user5722540 For all we know, you didn't solve anything, you conveniently preferred to ask for a solution, here. $\endgroup$
    – user436658
    Feb 15 '18 at 8:11
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Using integration by parts, let $$f(x)=\ln x\implies f'(x)=\frac1x$$ and $$g'(x)=1\implies g(x)=x$$ Then $$\int_0^1\ln x\,dx=[x\ln x]_0^1-\int_0^11\,dx=-1$$ Since each of the four quarters are identical, we have that the total area is $$I=4|-1|=\color{red}{4}$$

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