I am trying to find the volume of a solid located in the first octant and bounded by surfaces $z=x+y$, $xy=1$, $xy=2$, $y=x$, $y=2x$, $z=0$.
I set up the following integral:
$V = \int_{1/\sqrt 2}^1 \int_{1/x}^{2x} (x+y)dydx + \int_1^\sqrt 2 \int_x^{2/x} (x+y)dydx$.
I integrated it, checked it with Wolfram and it equals $\frac{4-\sqrt2}{3}$.
This is not the answer that I have in the textbook, but I believe the integration was correct as Wolfram gives the same result. So, I think either I set it up wrong or the answer in the textbook is wrong.
I also tried doing this integral using substitution. Setting $u=xy$, $v=y/x$, I got
$V=\int_1^2 \int_1^2 (\sqrt{u/v}+\sqrt{uv})|I| dvdu$, where
$I= v/2$.
This is the result of integration: https://www.wolframalpha.com/input/?i=int_1%5E2+int_1%5E2+(sqrt(x%2Fy)%2Bsqrt(x*y))*y%2F2+dydx
Again, when I integrate it, the result is the same as what I get with Wolfram. But it is not the same as using the first method or what is in the textbook. I am totally confused.