Use polar coordinates to find the volume of the given solid. Below the paraboloid z = 8 - 2x² - 2y² and above the xy-plane

Solution:

Given,

The paraboloid z = 8 - 2x² - 2y².

We can take it as,

x² + y² = 4.

r = 2 in polar coordinates

x = rcosφ, y = rsinφ

This is the circle whose center is the origin and its radius is R = 2.

To find the volume, you should calculate a double integral.

V = ∬(8 - 2r²)r dr dφ

Area in the xy plane 0 ≤ r ≤ 2.

The element of the area in polar coordinates is

dσ = r dr dφ

Reducing to definite integral,

V = 2π∫(8r -2r³) dr

V = 2π(4r² - 1/2 r⁴), r from 0 to 2

V = 16π.

Therefore, the volume of the given solid is 16π.

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Use polar coordinates to find the volume of the given solid. Below the paraboloid z = 8 - 2x² - 2y² and above the xy-plane

Summary:

Using polar coordinates the volume of the given solid is 16π. Below the paraboloid z = 8 - 2x² - 2y² and above the xy-plane.