Final answer:
To evaluate the improper integral ∫∫r2e−4(x2 y2) dx dy using polar coordinates, convert the integral to polar coordinates by substituting x = rcosθ and y = rsinθ. Simplify the expression, and then integrate with respect to r and θ.
Step-by-step explanation:
To evaluate the improper integral ∫∫r2e−4(x2 y2) dx dy using polar coordinates, we first need to convert the integral to polar coordinates. We know that in polar coordinates, x = rcosθ and y = rsinθ.
Substituting these into the integral, we have ∫∫r²e^(-4(r²cos²θ)(r²sin²θ))r dr dθ.
Simplifying the expression, we get ∫∫r^4e^(-4r^4cos²θsin²θ) dr dθ. Now we can evaluate this integral by multiplying the two results together and integrating with respect to r then θ.