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This is the first one I did which is just what we did in class:  https://www.geogebra.org/classic/afndqrc3
I was trying to get the integral, like I remember doing between functions last year, but the original equation wasn't a function, so I made it one: y=sqrt(x^2-1). I was able to find the area below the curve and then subtracted it from the whole triangular area to get the weird area we wanted (and multiplied by 2 because if I hadn't made it a function, the same area would be on the other side of the x-axis. This let me see the area at many different values for A, but I couldn't make the area negative. Here's my geogebra: https://www.geogebra.org/classic/ncy4g3dg
Third, I found the derivative of the hyperbola. I got dy/dx=x/y. I made a tangent line at A. My equation was y-y[A]=x[A]/y[A](x-x[A]). I don't really know what this tells me, but here's the geogebra: https://www.geogebra.org/classic/cp9s9ueu

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Here's my geogebra. I made the triangle like middy said. I wasn't really sure what to do about the other two sections but I kinda had a similar idea to saskia in that made you could use a function that is the top half of each branch, but I'm not really sure how. I am especially unsure about making it negative because even just the triangle area I made didn't go negative when I brought A below the X-Axis. https://www.geogebra.org/classic?lang=en . A question I have is whether Middy figured out how to do this himself or if he was shown how?

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Here's my graph of what we did in class. I'm not sure how to make the area negative. I was thinking if I move it below the x-axis that would work but it didn't seem to. Saskia went way in depth though and everything she did looks really good!

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Here's my Geogebra file:
https://www.geogebra.org/classic/tewnmxft
Like Victoria, I followed Middy's suggestion of seperating the are into three different regions, though I was only able to find the area of the triangular region.
I thought that I may be able to find the area by taking the integral of the function and multiplying by 2. However, I had trouble with this because I didn't convert it into a function.