Integrating with respect to area? Past paper question

[TheGreatCabbage]

This isn't exactly homework or coursework, it is a past paper question that I cannot find a solution to (my university doesn't like releasing answers for some reason unknown to me).
The question is attached as an image (edit: the image displays while editing but not in the post, so I'll try to type it out):

Question: Let A be the rectangle -2 ≤ x ≤ 2, -1 ≤ y ≤ 1. Evaluate ∫_A x³ dA.

I attempted to solve this by parametising x and y in terms of a new variable t, such that x = 2t and y = t for -1 ≤ t ≤ 1. Then I used the fact that A = xy to obtain A = xy = 2t² and dA = 4t dt. Substituting into the integral with limits -1 and 1 gave my a final answer of 64/5, but I lack confidence in my method and answer.
Should I use a double integral? Any help would be much appreciated.

 

[Mark44]

This isn't exactly homework or coursework, it is a past paper question that I cannot find a solution to (my university doesn't like releasing answers for some reason unknown to me).
The question is attached as an image (edit: the image displays while editing but not in the post, so I'll try to type it out):

Question: Let A be the rectangle -2 ≤ x ≤ 2, -1 ≤ y ≤ 1. Evaluate ∫_A x³ dA.

I attempted to solve this by parametising x and y in terms of a new variable t, such that x = 2t and y = t for -1 ≤ t ≤ 1. Then I used the fact that A = xy to obtain A = xy = 2t² and dA = 4t dt. Substituting into the integral with limits -1 and 1 gave my a final answer of 64/5, but I lack confidence in my method and answer.
Should I use a double integral? Any help would be much appreciated.

I don't see any reason NOT to use a double integral.

BTW, what would you get from this integral? $\int_{-1}^1 x^3 dx$

 

[TheGreatCabbage]

I don't see any reason NOT to use a double integral.

BTW, what would you get from this integral? $\int_{-1}^1 x^3 dx$

Zero :)
I think I understand it now, I tried a double integral method earlier and it gave me zero and I thought that it was wrong, but after considering it properly I think that it's right.