Questions: Use polar coordinates to find the volume of the given solid below the paraboloid (z=18-2 x^2-2 y^2) and above the (xy)-plane.

Transcript text: Use polar coordinates to find the volume of the given solid below the paraboloid $z=18-2 x^{2}-2 y^{2}$ and above the $x y$-plane

Solution

Solution Steps

To find the volume of the solid below the paraboloid $ z = 18 - 2x^2 - 2y^2 $ and above the $ xy $-plane using polar coordinates, we need to:

Convert the given equation to polar coordinates.
Set up the double integral in polar coordinates.
Evaluate the integral to find the volume.

Step 1: Convert to Polar Coordinates

The given paraboloid is $ z = 18 - 2x^2 - 2y^2 $. In polar coordinates, we substitute $ x = r \cos(\theta) $ and $ y = r \sin(\theta) $. This transforms the equation to: \[ z = 18 - 2r^2 \]

Step 2: Set Up the Double Integral

To find the volume $ V $ of the solid below the paraboloid and above the $ xy $-plane, we set up the double integral in polar coordinates: \[ V = \int_0^{2\pi} \int_0^{\sqrt{9}} (18 - 2r^2) r \, dr \, d\theta \] Here, the limits for $ r $ are from $ 0 $ to $ \sqrt{9} $ (where $ z = 0 $), and the limits for $ \theta $ are from $ 0 $ to $ 2\pi $.

Step 3: Evaluate the Integral

Evaluating the integral gives: \[ V = 81\pi \] This represents the volume of the solid.