Final answer:
To compute the volume of a solid under 2x+y+z=4 and above the disk x²+y² ≤1, we express x and y in polar coordinates and integrate the resulting function z over the specified domain.
Step-by-step explanation:
The question asks to find the volume of a solid defined by a specific plane and a disk using polar coordinates. The plane equation given is 2x+y+z=4, and the disk is restricted by x²+y² ≤1. To find this volume we can integrate the function that represents the height of the plane above the disk in polar coordinates. In polar coordinates, x=rcosθ and y=rsinθ, where r is the radius and θ is the angle. The plane's equation becomes 2rcosθ+rsinθ+z=4. Solving for z, we get z=4-2rcosθ-rsinθ. As the disk has a radius of 1, we integrate over r from 0 to 1 and θ from 0 to 2π to obtain the volume.
The integration setup is therefore:
∫∫^{2π}_{0}∫^{1}_{0} (4-2rcosθ-rsinθ) r dr dθ
To solve, we multiply the integrand by r (the Jacobian when converting to polar coordinates) and integrate with respect to r first, then θ. After calculating both integrals, the evaluated expressions will yield the volume of the solid.