In the $xy$-plane a function is given: $f(x,y)=x+y$. Let $A$ be the area that is in the $xy$-plane and is encapsulated by $x=0$, $x=\frac{\pi}{4}$, $y=x+\cos(x)$ and $y=x+\sin(x)$.
a) Make a parametrization of $A$ and find its jacobi-function. Then calculate the area.
In my textbook the parametrization i am to make is on the form: $r(u,v)=(x(u,v),y(u,v))\in \mathbb{R}^2$, $u\in [a,b]$,$v\in [c,d]$
I believe my parametrization is wrong. this is what i did:
since the two $x$-values are vertical lines they they are given the parameter $u$ and can be described like this: $x=u, u\in [0,\frac{\pi}{4}]$
The horizontal lines are defined by $y=x+\cos(x)$ and $y=x+\sin(x)$ and are given the parameter $v$. I have given this parameter $v$ the interval $[0,1]$ because when i combine my functions: $(1-v)(u+\cos(u))+v(u+\sin(u))$, then whenever $v=0$ only $u+\cos(u)$ is true. And when $v=1$, $u+\sin(u)$ is true.
So my parametrization becomes like this: $$r(u,v)=(u,(1-v)(u+\cos(u))+v(u+\sin(u)))$$
I then move on to calculate the jacobian, where the first step is to take the partial derivative and i get a zero-matrix: $\begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}$. Which can't be right given I have to calculate the area in the end.
What am I doing wrong?