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The motivation behind my question is the exercise "find the area between $x + y = 0$ and $x = y^2 + 3y$". The typical approach taught in order to solve this question is to integrate with respect to y. However, because I'm less familiar graphing with respect to y than with respect to x, is it not equally valid to rewrite the exercise, interchanging x and y, as "find the area between $y + x = 0$ and $y = x^2 + 3y$" and solving with respect to x? I've solved this particular case both ways and gotten agreeing answers of $\frac{32}{3}$, but I'd like to know if the approach holds true in general.

(My intuition says yes, it must hold true in general, because in the problem definition x and y are just labels. They could be $\alpha$ and $\beta$ or any other pair of symbols and the problem would remain the same.)

Scott Emmons
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Your intuition is correct. Swapping $x$ and $y$ mirrors your function across the line $y = x$, which preserves area.

6c1
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    An explanation of why swapping $x$ and $y$ mirrors over $y = x$ can be found here: http://math.stackexchange.com/a/741376/191537 – Scott Emmons Nov 29 '15 at 01:43