need help with a change of variable question

Question

For the following question, I am wondering if someone can give me some assistance:

Evaluate the integral

$$\int_{0}^{1} \int_{0}^{x} \sqrt{{x}^{2}+{y}^{2}} \,dy\,dx$$

using the transformation $\ x=u, y=uv\ $

I plugged into the integrand $u$ for $x$ and $uv$ for $y$ with the jacobian of the derivative matrix of $(u, uv)$ being $u$. I tried various transformations that maps $(0,0), (0,1), (1,1)$ to $(u, uv)$ like

$$\int_{0}^{1} \int_{v}^{1} u\sqrt{u^{2}({v}^{2}+1)}\ dv\ du$$ or $$\int_{0}^{1} \int_{0}^{\frac{1} {v}} u\sqrt{u^{2}({v}^{2}+1)}\ du\ dv$$

But I keep getting the wrong answer from$\int_{0}^{1} \int_{0}^{x} \sqrt{{x}^{2}+{y}^{2}} \,dy\,dx.$ I am not sure what I am missing here. If someone can point me to the right direction, it will be much appreciated. Thank you in advance.

Answer 1

$$\int_{0}^{1}\int_{0}^{x}\sqrt{x^2+y^2}\,dy\,dx = \int_{0}^{1}\int_{0}^{1}x\sqrt{x^2+x^2 z^2}\,dz\,dx = \int_{0}^{1}x^2\,dx\int_{0}^{1}\sqrt{1+z^2}\,dz$$ by the change of variable $y\mapsto xz$ and Fubini's theorem. The RHS equals $$ \frac{1}{3}\left\{\left[z\sqrt{1+z^2}\right]_{0}^{1}+\int_{0}^{1}\frac{z^2\,dz}{\sqrt{1+z^2}}\right\}=\frac{1}{6}\left[\sqrt{2}+\text{arcsinh}(1)\right]$$ by integration by parts and writing $z^2$ as $(1+z^2)-1$.

Answer 2

To see how the domain changes with the change of variable we can notice that $u=x$ and $v=\frac{y}{x}$.

So $(u,v)$ is in the square $[0,1] \times [0,1]$.

More precisely with $\phi: (u,v) \mapsto (u,uv)$ you have to check that $\phi([0,1] \times [0,1])=\{0 \leq y \leq x \leq 1\}$.