If there are two double integrals and they have the same size, is there a rule that supports combining them?
For example: $$\int_{-1}^{1} \int_{-1}^{x} \left(1 + x\sin^2(y) + y^3\cos^4(x) \right)\, dy\, dx + \int_{-1}^{1} \int_{-1}^{y} \left(2 + x\sin^2(y) + y^3\cos^4(x) \right) \,dy \,dx $$ in the image's below, you can see they take of the same space but the limits are different. How is this solved?
limits graph:
Yes, you can merge the domains of integration: if $f$ is continuous in $[-1,1]\times [-1,1]$ then $$\underbrace{\int_{x=-1}^{1} \int_{y=-1}^{x}}_{\text{triangle $(-1,-1)$, $(1,-1)$, $(1,1)$}}\!\!\!\!\!\!\!\!\!\!\!\! f(x,y)dx + \underbrace{\int_{y=-1}^{1} \int_{x=-1}^{y}}_{\text{triangle $(-1,-1)$, $(-1,1)$, $(1,1)$}} \!\!\!\!\!\!\!\!\!\!\!\! f(x,y) dx dy$$ is equal to $$\underbrace{\int_{x=-1}^1\int_{y=-1}^1}_{\text{square $(-1,-1)$, $(1,-1)$, $(1,1)$, $(-1,1)$}}\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! f(x,y) dxdy.$$ Note that your $f(x,y)$ is a sum of products $g(x)\cdot h(y)$, therefore to evaluate the double integral you may use linearity and $$\int_{x=-1}^1\int_{y=-1}^1 g(x)\cdot h(y) dxdy=\left(\int_{x=-1}^1 g(x) dx\right)\cdot \left(\int_{y=-1}^1 h(y) dy\right).$$ Finally recall that the integral of an odd function over $[-1,1]$ is zero.