Confused about this solution manual approach-link included

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Choosing dq in different quadrants is valid, but it will affect the signs in calculations. Regardless of the quadrant chosen, the integration should still be performed from 0 to 2 pi, encompassing the entire circle. This approach ensures that the final answer remains consistent. The key is to maintain a clear understanding of the coordinate system and its implications on the signs. Ultimately, the method yields the same result despite the initial quadrant selection.
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Yes, you should be able to draw the diagram in any quadrant. But you will still integrate from 0 to 2 pi (around the whole circle), so you'll get the same answer.
 

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