Final answer:
To evaluate the integral of the region bounded by the semicircle x = 81 - y^2 and the y-axis in polar coordinates, convert x and y to their polar forms and find the limits for r and θ. The integral becomes a product of a radial and an angular integral that can be separately evaluated.
Step-by-step explanation:
To evaluate the integral ∫∫D |e^{-x^2 - y^2} dA, where D is the region bounded by the semicircle x = 81 - y^2 and the y-axis, we must convert the equation to polar coordinates. In polar coordinates, x = r × cos(θ) and y = r × sin(θ), and the differential area element dA becomes rdrdθ. The semicircle in polar coordinates is characterized by a constant radius, r, from 0 to 9 (since the original x boundary is a semicircle with radius 9), and an angle, θ, from 0 to π (half the circle). The integral then simplifies to ∫ π 0∫ 9 0 e^{-r^2} r drdθ, which can be evaluated as a product of a radial integral and an angular integral.