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-

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.

[That is because you would then be calculating the surface area over the square .]
The solution book says to convert the function to polar coordinates as such -

,

[Correct except that should be .]
I'm unsure of the bounds for theta, wouldn't 0-2\pi give the surface area of the whole sphere

[It's not a sphere, it's a paraboloid, and you're calculating the area of that part of its surface that lies above the disk , which in polar coordinates corresponds to .] 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?