- #1
Math11
- 1
- 0
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
Last edited: