I am having trouble calculating this integral.
Let $f:\mathbb{R}^2\rightarrow \mathbb{R}$ defined by \begin{equation} f(x,y) := \begin{cases} 1 & \text{if} \ x\geq0, \ x\leq y \leq 1+x,\\ -1 &\text{if} \ x\geq0, \ 1+x\leq y \leq 2+x, \\ 0 &\text{else}. \end{cases} \end{equation} Calculate \begin{equation} \int_\mathbb{R} \bigg(\int_\mathbb{R}f(x,y)\text{d}\lambda_1(y)\bigg)\text{d}\lambda_1(x) \ \ \ \text{and} \ \ \int_\mathbb{R} \bigg(\int_\mathbb{R}f(x,y)\text{d}\lambda_1(x)\bigg)\text{d}\lambda_1(y). \end{equation}