Final answer:
The total charge on the disk is found by integrating the given charge density function over the entire area of the disk using polar coordinates. The integration process involves converting the charge density into polar form and then integrating with respect to the radial and angular coordinates.
Step-by-step explanation:
The task is to find the total electric charge on a disk with a radius of 2, where the charge density ρ(x, y) is given by ρ(x, y) = 4x + 4y + 4x² + 4y². To find the total charge, we integrate the charge density over the area of the disk.
Integration Process
The integration will be performed in polar coordinates for simplicity, as the charge density is symmetrically distributed over the disk. We use r for the radial distance from the origin and θ for the angular coordinate.
The charge density function ρ(r, θ) in polar coordinates is ρ(r, θ) = 4r cos(θ) + 4r sin(θ) + 4r². The total charge Q is the double integral of the charge density over the disk's area:
Q = ∫∫_D ρ(x, y) dA
To compute this, we integrate from r = 0 to 2 and from θ = 0 to 2π:
Q = ∫_0^{2π} ∫_0^2 (4r cos(θ) + 4r sin(θ) + 4r²) r dr dθ
Upon evaluating this integral, we obtain the total charge on the disk.