Question

Use polar coordinates to find the volume of the solid below the paraboloid z=72−2x ^2 −2y ^2 and above the xy-plane.

Answer 1

To find the volume of the solid below the paraboloid z = 72 - 2x^2 - 2y^2 and above the xy-plane using polar coordinates, set up a triple integral and carry out the integration to calculate the volume.

Step-by-step explanation:

To find the volume of the solid below the paraboloid z = 72 - 2x^2 - 2y^2 and above the xy-plane using polar coordinates, we can set up a triple integral.

First, we need to express the equation of the paraboloid in terms of polar coordinates. We know that x = r cos(theta) and y = r sin(theta). Substituting these values into the equation, we get z = 72 - 2(r cos(theta))^2 - 2(r sin(theta))^2.

The bounds of integration for the triple integral are:
- r ranges from 0 to √(36/2) (since the paraboloid is symmetric, we only need to consider the positive hemisphere)
- theta ranges from 0 to 2π (full revolution around the z-axis)

The volume is given by the triple integral:
Volume = ∫∫∫ r dz dr dtheta. By carrying out the integration, the volume can be calculated.