Final answer:
To evaluate the given improper integral in polar coordinates, substitute the polar coordinates x = rcos(θ) and y = rsin(θ) into the integral and determine the bounds for the integration based on the third quadrant.
Step-by-step explanation:
To evaluate the given improper integral in polar coordinates, we need to express the integral in terms of polar coordinates and determine the bounds for the integration.
First, we can express the integral as:
∫∫D 1/(1+x²+y²)³ dA
Substituting the polar coordinates x = rcos(θ) and y = rsin(θ) into the integral, we get:
∫∫D 1/(1+r²)dA
To determine the bounds for the integration, we note that D is the third quadrant, which corresponds to θ values between 180° and 270°. The radial limits r can be determined by considering the region in the third quadrant.
We evaluate the integral by solving the limits of the integration and carrying out the integration.