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PostPosted: Sat, 19 May 2012 16:58:46 UTC 
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Question: Find the area of the following bounded region:
{ (x,y) | 1 ≤ x ≤ \sqrt{2} and 0 ≤ y ≤ ln(x + \sqrt{x^2 - 1} }


http://i1084.photobucket.com/albums/j40 ... MG007a.png
http://i1084.photobucket.com/albums/j40 ... MG007b.png

I get ln|1-1| near the end...


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PostPosted: Sat, 19 May 2012 22:53:43 UTC 
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A-R-Q wrote:
Question: Find the area of the following bounded region:
{ (x,y) | 1 ≤ x ≤ \sqrt{2} and 0 ≤ y ≤ ln(x + \sqrt{x^2 - 1} }


http://i1084.photobucket.com/albums/j40 ... MG007a.png
http://i1084.photobucket.com/albums/j40 ... MG007b.png

I get ln|1-1| near the end...


As an FYI, a lot of people will ignore posts that are hard to read (for example, I'm about 30/70 will/won't read ratio for things that I have to rewrite on my own to understand). If you want to be more likely to receive help you really should learn a little TeX, you're at 411 posts on a place where that is the standard.

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PostPosted: Sun, 20 May 2012 09:31:35 UTC 
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A-R-Q wrote:
Question: Find the area of the following bounded region:
{ (x,y) | 1 ≤ x ≤ \sqrt{2} and 0 ≤ y ≤ ln(x + \sqrt{x^2 - 1} }


http://i1084.photobucket.com/albums/j40 ... MG007a.png
http://i1084.photobucket.com/albums/j40 ... MG007b.png

I get ln|1-1| near the end...


How did you get x^2-1 in the denominator instead of \sqrt{x^2-1}?

Integrate by parts gives
\displaystyle\int_1^x\cosh^{-1}(\xi)\,\mathrm{d}\xi=x\cosh^{-1}(x)-\underbrace{\sinh(\cosh^{-1}(x))}_{\sqrt{x^2-1}}

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\begin{aligned}
Spin(1)&=O(1)=\mathbb{Z}/2&\quad&\text{and}\\
Spin(2)&=U(1)=SO(2)&&\text{are obvious}\\
Spin(3)&=Sp(1)=SU(2)&&\text{by }q\mapsto(\mathop{\mathrm{Im}}\mathbb{H}\ni p\mapsto qp\bar{q})\\
Spin(4)&=Sp(1)\times Sp(1)&&\text{by }(q_1,q_2)\mapsto(\mathbb{H}\ni p\mapsto q_1p\bar{q_2})\\
Spin(5)&=Sp(2)&&\text{by }\mathbb{HP}^1\cong S^4_{round}\hookrightarrow\mathbb{R}^5\\
Spin(6)&=SU(4)&&\text{by the irrep }\Lambda_+\mathbb{C}^4
\end{aligned}


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