Final answer:
The question involves evaluating an iterated integral by converting to polar coordinates. The integral is over the area inside a circle in Cartesian coordinates and simplifies in polar coordinates, with the limits 0 to 9 for r and 0 to 2π for θ.
Step-by-step explanation:
The student is asking how to evaluate an iterated integral by converting from Cartesian to polar coordinates. The boundaries of integration in Cartesian coordinates appear to be the circle x2 + y2 <= 81, and the function to be integrated is e-(x2 + y2). When converting to polar coordinates, x = r*cos(θ), y = r*sin(θ), and the differential area element dxdy becomes r dr dθ. The function to be integrated simplifies to e-r2, and the new limits of integration become 0 to 2π for θ and 0 to 9 for r.
To evaluate the integral, one must integrate e-r2 with respect to r from 0 to 9, and then integrate the result with respect to θ from 0 to 2π. Adding the results from both integrations gives the final answer to the iterated integral.