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Integration involving Hyperbolic Functions Problem

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Hi, could anybody help me with the following question? I can not figure it out how to find the same constant of proportionality for the rectangular hyperbola. I have attached the book's question and solution as well as my working....... I would be really grateful if anyone can help.
 

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PlanetMaster

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For area of the sector \(OAP_{1} \), we can simply use
\(\text{Area of a sector}=\frac{1}{2}{r^{2}}\theta \) and since \(r=a \) (from \(A \) coordinates), we have
\(\text{Area of }OAP_{1}=\frac{1}{2}{a^{2}}\theta \)

So, area of \(OAP_{1} \) is proportional to \(\theta \).

For the area of \(OAP_{2} \), I'd split the diagram up like this to make life easier
d8gtwtg5w3.png
Now we have
\(\text{Area of }OAP_{2}=\text{Area of }OAT+\text{Area of }ATP_{2} \)

We'll need the equation of line \(OP_{2} \). Since its a straight line,
\(y=mx+c \)
\(m=\frac{asinh{\phi}}{acosh{\phi}}=\tanh{\phi} \)
\(c=0 \)
(as the line is extending from origin)
So equation of straight line \(OP_{2} \) is
\(y=xtanh{\phi} \)

Now we can start finding areas.

Since OAT is a right-angled triangle, its area
\(=\frac{1}{2}\times{base}\times{height} \)
\(=\frac{1}{2}\times{a}\times{atanh{\phi}} \)
(from the straight line equation)
\(=\frac{1}{2}{a^2}{tanh{\phi}} \)

\(\text{Area of }ATP_{2}=[\text{Area of }SATP_{2}-\text{Area of }SAP_{2}] \)
\(=\int_{a}^{acosh{\phi}}({xtanh{\phi}-\sqrt{{x^2}-{a^2}}})\mathop{dx} \)
\(=\frac{1}{2}{a^{2}}({\phi}-tanh{\phi}) \)


So finally,
\(\text{Area of }OAP_{2}=\text{Area of }OAT+\text{Area of }ATP_{2} \)
\(=\frac{1}{2}{a^2}{tanh{\phi}}+\frac{1}{2}{a^{2}}({\phi}-tanh{\phi}) \)
\(=\frac{1}{2}{a^{2}}{\phi} \)


So, area of \(OAP_{2} \) is proportional to \(\phi \).

And \(OAP_{1}\text{ and }OAP_{2}\) have the same constant of proportionality i.e. \(\frac{1}{2}a^{2} \).

Hope this helps!
 
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I did it this way but then realized that I have substituted the wrong limit. Then after rectifying it, I reached the desired conclusion. I have attached my steps.... Thanks for your help PlanetMaster
 

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