Final answer:
The integral ∫∫R x²y dA over the top half of a disk in polar coordinates involves converting the Cartesian expression to polar form, setting up the integral bounds for r and θ, and using the polar area element r dr dθ to compute the final answer.
Step-by-step explanation:
The task involves computing the integral of x²y over the region R, where R is the top half of the disk centered at the origin with radius 3. To solve this, we convert the Cartesian coordinates to polar coordinates. In polar coordinates, the integral ∫∫R x²y dA is transformed into an integral with new limits reflecting the shape and extent of R in polar terms.
To perform this integration in polar coordinates, we consider that x = r cos(θ) and y = r sin(θ), resulting in the new integrand r³ cos(θ)² sin(θ). We then integrate over the appropriate bounds for r and θ. The limits for r are from 0 to 3, because the disk has a radius of 3, and the limits for θ are from 0 to π, since we're only considering the top half of the disk.
The differential area element in polar coordinates is given by dA = r dr dθ. The integral, therefore, becomes a double integral where we first integrate with respect to θ from 0 to π and then with respect to r from 0 to 3. Combine these elements, and we achieve the final answer.