2 to the negative 3rd power represented another way
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Yes it's exactly due to symmetry. Specifically, symmetry about the x axis.
Simply writing , without the 2 out front, will only get you the area of the portion shown in red (see diagram below).
The blue region has equal area of the red region due to symmetry.
So,
represents the red and blue regions combined.
The portion in green is which is the integral of the difference of the upper and lower curves over the interval
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In all honesty, it's probably easier to integrate with respect to y since the given functions are in terms of y initially. Also, there isn't a junction point in which the curves swap places in terms of which one is larger. However, it doesn't hurt to have practice in integrating with respect to x.
If you're curious about what the y integral looks like, then it would be
You can use a tool like WolframAlpha to check that both integral expressions result in 36 to help confirm that they represent the same overall area (just in different ways of course).
