I just started seeing integrals in Calc 2 and I have a question about limits of integration. For example, let's say I have $x = \pm 1$ and $y = x$. Would the region delimited by these lines be the right triangle on the positive side and the upside down right triangle on the negative side? I ask this because most of what I've seen so far deals with curves creating one single region and it is very straightforward to see, but this one seems to be generating two separate regions. In this case, should I write $R = \{(x,y) \in \mathbb{R}^2\,|\,-1 \leq x \leq 1\,\,\,\text{and}\,\,\,0 \leq y \leq x\,\,\,\text{and}\,\,\,0 \geq y \geq x\}$ and let the integrals be
$$\int \int_R f = \int_{0}^{1}\int_{0}^{x} f(x,y) dydx + \int_{-1}^{0}\int_{x}^{0} f(x,y)dydx$$
or write $R = \{(x,y) \in \mathbb{R}^2\,|\,0 \leq x \leq 1\,\,\,\text{and}\,\,\,0 \leq y \leq x\}$ and just multiply the integrals by 2, like this
$$\int \int_R f = 2\int_{0}^{1}\int_{0}^{x} f(x,y) dydx\,\,?$$
It is also possible none of these regions are correct. I'm a bit confused about this.
The way you have written $R$ initially is not correct, i.e. it does not match your description. I would suggest $$R = \{(x,y)\in \mathbb R^2: -1\le x\le 1, y < x \text{ if } x > 0, y > x \text{ if } x \le 0\}$$ Also, you cannot just write $$R = \{(x,y)\in \mathbb R^2: 0\le x\le 1, y < x\}$$ and multiply the integral by $2$, because you do not know enough about $f(x,y)$. What if $f(x,y) = 0$ for $x < 0$, but $f(x,y)\ne 0$ for $x \ge 0$? Certainly this does not work. I leave it to you to see under what conditions on $f$ can you consider $R$ as above and multiply the integral by $2$.