Finding Volume over Circle x²+y² ≤ 4 under f(x,y) = 4-x²-y²

Calculus 3 · Axiom Academy

EXAMPLE Finding Volume over under f(x,y)=4-x^2-y^2 Convert to polar coordinates and evaluate a double integral to find the volume under a paraboloid. Find the volume of the solid bounded above by the surface f(x,y) = 4 - x^2 - y^2 and below by the disk in the xy -plane. Cross-section in the r – z plane. The surface z = 4 - x^2 - y^2 becomes z = 4 - r^2 ; it caps a dome that meets z=0 at r=2 . For a fixed r , the height is 4 - r^2 — this is what we integrate over the disk . Excellent work! You found the volume using polar coordinates. Here's what carried the solution: Recognize circular symmetry: when the region is , polar coordinates greatly simplify the problem. Polar substitution: replace x^2 + y^2 with r^2 . The surface f(x,y) = 4 - x^2 - y^2 became . Identify the bounds carefully: for a disk of radius 2, and (full rotation). Never forget the Jacobian: in polar coordinates . That extra r factor is crucial. This technique is powerful for any region with circular or rotational symmetry — you'll reuse it constantly in Calculus 3 for volumes, surface areas, and centers of mass.

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