Final answer:
To express the given integral in polar coordinates, we perform a variable substitution where x = r cosθ, y = r sinθ, and include the Jacobian r in the integral. The polar form of the integral is the double integral from 0 to 2π in θ and 0 to 1 in r with the integrand being r^2/(1+r^2).
Step-by-step explanation:
The given integral in Cartesian coordinates is −1∫1 ₀0∫√1−y² 1/ (1+x²+y²) dxdy, which describes the integration over a circular region. To express this integral in polar coordinates, we substitute x = r cosθ and y = r sinθ, where r is the radius from the origin to a point in the region and θ is the angle from the positive x-axis to the point. The bounds for r will be from 0 to 1, as it's the radius of a unit circle, and θ will range from 0 to 2π, covering the entire circle. Also, remember to include the Jacobian determinant, which is r in polar coordinates, when changing variables to polar coordinates. The integral in polar form becomes:
₀0∫2π ₀0∫1 ²/(1+r²) r dr dθ