Double Integral over general region

[Math11]

Homework Statement

Find the integral using a geometric argument.

∫∫_D√(16 - x² - y²)dA
over the region D where D = {(x,y) : x² + y² ≤ 16}

By the way, the subscript D next to the integral refers to the region over which the function is integrated.

Homework Equations

∫∫f(x,y)dxdy = ∫∫f(x,y)dydx

Don't think it's a polar coordinate question but here is the equation for a double integral in polar coordinates anyway:
∫∫_Df(x,y)r dA

The Attempt at a Solution

Attempted to solve the equation using a double integral in polar coordinates with:
0 ≤ r ≤ 4
0 ≤ θ ≤ pi
got 64π/3 but I'm not certain whether it is correct. I'm not sure whether the graph approaches the x-y plane asymptotically or not. The equation for f(x,y) = √(16 - x² - y²) describes what seems like the positive z half of a sphere

Edit: Made a second attempt. Assumed the shape to be the top half of a sphere so the equation became:
(0.5)∫∫_D(16 - x² - y²)dA
Where x ranges from 0 to 4 and y ranges from -√(16-x²) to √(16-x²)

Any help would be much appreciated.
Thank you

 

[STEMucator]

Draw your region, its a circle with radius 4 with a center at (0,0).

There are four curves you need to consider when setting up your integral. You can find these curves by solving x² + y² = 16.