Double integral surface area with polar coordinates

Problem: Find the surface area of the paraboloid that lies above the xy-plane. Finding where the function crosses the xy plane- taking the gradient- Useing the surface area formula- <br> <br><br>This is where I'm unsure of how to continue. It seems that I could just set my bounds to be [-2,2] and [0,2] but this gives and incorrect answer.

The solution book says to convert the function to polar coordinates as such -

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,<br><br>I'm unsure of the bounds for theta, wouldn't 0-2\pi give the surface area of the whole sphere and not the area above ? How would you compute the total surface area if this equation just gives it to you above the xy-plane?<br>