Final answer:
To evaluate the given integral using polar coordinates, express the differential area element in polar coordinates and evaluate the resulting double integral by integrating with respect to r and θ.
Step-by-step explanation:
To evaluate the integral, ∫∫rsin(x^2 y^2)dA, using polar coordinates, we need to express the differential area element, dA, in terms of polar coordinates. In polar coordinates, dA is given by dA = r dr dθ. Here, 4 ≤ x^2 + y^2 ≤ 49 represents the region in polar coordinates.
We can express x^2 + y^2 as r^2. So, the integral becomes ∫∫r sin(r^2)dA. Replacing dA with r dr dθ, the integral becomes ∫∫r^2 sin(r^2) dr dθ.
We first integrate with respect to r from 0 to the upper limit, which is √49, and then integrate with respect to θ from 0 to 2π, which covers the entire region in polar coordinates. By evaluating these integrals, we can find the final result.