This is from Gamelin's Complex Analysis, I am an in introductory course to complex analysis and am having trouble understanding a particular solution to this problem.
The question: Let $f(z)$ be a bounded analytic function, defined on a bounded domain $D$ in the complex plane, and suppose that $f(z)$ is one-to-one. Show that the area of $f(D)$ is given by $$\mathrm{Area}\,(f(D))=\iint_\limits D |f'(z)|^2dxdy$$
What I understand from my own lecture notes which I apply to this question is: $f:D\longrightarrow\mathbb{C}$. Given that $f$ is analytic, defining $f$ to be $f(z)=u(x,y)+iv(x,y)$ the real valued functions $u,v: D\longrightarrow \mathbb{R}$ have continuous partial derivatives that which satisfy the Cauchy-Riemann conditions. This gives us that the determinant of the jacobian is $|f'(z)|^2$ which is constant (and I see that it is part of the integral).
Also, I understand that $|f(z)| \leq M$ for some $M< \infty$, but I can only reason that this assumption is made so we can actually integrate the function. But the one-to-one assumption is giving me no progress in terms of what to do, unless it is meant for the change of variables to be well defined?
If someone can help me understand this solution or just expand on it. It has been a couple years since I have seen multi-variable as I am a non-traditional student so I maybe it is the Jacobian and change of variables I am having trouble with
The particular solution:
Given $f:D\longrightarrow\mathbb{C}$, with $|f(z)| \leq M$ for some $M< \infty$, because $f$ is assumed to be one-to-one, we may compute
$$\underbrace{\mathrm{Area}\,(f(D))=\iint\limits_{f(D)}^{}dudv}_{\text{why?}}$$ By change of variables $(u(x,y),v(x,y)) \mapsto (x,y)$ and $\det[J(f)]= |f'(z)|^2$ we get
$$\mathrm{Area}\,(f(D))=\iint_\limits D |f'(z)|^2dxdy$$