Volume Under a Paraboloid23456

Find the volume under the surface $z = 9 - x^2 - y^2$ and above the $xy$-plane.

This problem involves finding the volume of a solid region bounded above by a paraboloid surface and below by the xy-plane. The given surface equation is a standard form of an elliptic paraboloid, and the task is to calculate the volume under this paraboloid using integration techniques. Here, the surface is expressed as z equals nine minus the sum of the squares of x and y, a common form encountered in multivariable calculus.

To tackle this problem, one effective method is to employ double integration over a circular region. The geometry suggests that the limits of integration will be circular, making it intuitive to switch to polar coordinates, where x is represented as r times the cosine of theta and y as r times the sine of theta. This transformation simplifies the integration process as the bounds become more manageable, matching the circular symmetry of the paraboloid.

Emphasizing the strategy, understanding the choice of coordinate transformation and the setup of the integrals is crucial. The integrand evolves naturally from the expression of z, and the bounds for r and theta are derived from the circular footprint of the surface on the xy-plane. This problem reinforces the powerful technique of coordinate changes in evaluating double integrals, particularly when symmetry is present. Additionally, it serves as a practical application of interpreting geometrical relationships and using calculus tools to find volumes in multivariable contexts.