Final answer:
The integral ∫∫ (R) √(4-x²-y²) dA over the region R of a semicircle can be evaluated using polar coordinates. The r variable ranges from 0 to 2, and θ from 0 to π/2. The area element in polar coordinates is represented by dA = rdrdθ.
Step-by-step explanation:
The question asks to evaluate the double integral ∫∫ (R) √(4-x²-y²) dA, where the region R is defined by the set of points (x, y) such that x² + y² ≤ 4, with the additional constraint that x ≥ 0. This describes a semicircle with radius 2 centered at the origin in the first quadrant of the xy-plane.
To solve the integral, it is beneficial to use polar coordinates where x = rcos(θ) and y = rsin(θ). The Jacobian determinant when converting to polar coordinates is r, which accounts for the area element in polar coordinates, dA = rdrdθ.
To set up the integral in polar coordinates, we take into account that r ranges from 0 to 2 (the radius of the semicircle) and θ ranges from 0 to π/2 (since we are only considering the first quadrant). Therefore, our double integral becomes ∫ from 0 to π/2 ∫ from 0 to 2 √(4-r²)r dr dθ. To perform the integral, we integrate with respect to r first, followed by θ.
The integral of √(4-r²)r with respect to r can be recognized as an integral suitable for a trigonometric substitution, or by observing a table of integrals. After evaluating the integral with respect to r, and then with respect to θ, we obtain the final answer.
The process involves computational steps such as substituting limits of integration and simplifying the expressions obtained after the integration.