Evaluate $\int_{-1}^{1}\int_{x}^{2x-1}dydx$ 
$$\int_{-1}^{1}\int_{x}^{2x-1}dydx$$

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My attempt:
$$I_1:=\int_{-1}^{1}\int_{x}^{2x-1}dydx$$
$$=\int_{-1}^{1}\bigg[\int_{x}^{2x-1}dy\bigg]dx$$
$$=\int_{-1}^{1}\bigg(x-1\bigg)dx=\boxed{\color{red}{-2}}$$
Now I need to evaluate with changing the limis:
$$I_2:=\int_{-3}^{-1}\int_{-1}^{\frac{y+1}{2}}dxdy+\int_{-1}^{1}\int_{y}^{\frac{y+1}{2}}dxdy$$

1) Is it ok that before changing the limits the area is negative?
2) The new limits that I have done are correct?


*

*Your first integral should be
$$\int_{-1}^t \int_{2x-1}^{x}dy\,dx$$


because $2x - 1 < x$ for $x \in [-1,1]$
With that, the result is positive


*Your second integral is correct

