Area of a Parametrized Surface

[Differentiate1]

Here's my work: http://i.imgur.com/UMj72Ub.png

I used the surface area differential for a parametrized surface to solve for the area of that paraboloid surface. My friend tried solving this by parametrizing with x and y instead of r and theta which gave him the same answer. I would greatly appreciate it if anyone else can verify if this answer is correct as it looks out of the ordinary.

Thanks,

Differentiate1

 

[haruspex]

You've a few typos in there, but it is essentially right.
Some spurious 'r' factors in r_r. You don't want the modulus signs around the first mention of r_r x r_θ.
You should mention the reasoning behind writing z = r² instead of z = 4-r².

 

[Differentiate1]

You've a few typos in there, but it is essentially right.
Some spurious 'r' factors in r_r. You don't want the modulus signs around the first mention of r_r x r_θ.
You should mention the reasoning behind writing z = r² instead of z = 4-r².

Sorry about the part where I wrote z = r². I essentially set z = 0 and moved the 4 over and got -4 = -x²-y². Eliminate the negatives and I ended up with 4 = x²+y² where r = 2. Hence why I set z = r².

Thanks again!