Final answer:
The integration region is a semi-circle centered at (4,0) with a radius of 4, converted to polar coordinates with θ from 0 to π and r from 0 to 8cosθ. The polar integral is evaluated to find the solution.
Step-by-step explanation:
The student asks to first sketch the region of integration and then evaluate the integral ∨⁸₄ ∨ƒ⁸⁰ 1/√x²+y² dydx, where f(x) = √8x-x², by changing to polar coordinates. The given function f(x) represents a semi-circle when rearranged as x² + y² = 8x, which can be rewritten as (x-4)² + y² = 4², centering the circle at (4,0) with a radius of 4. By changing to polar coordinates, the limits for θ will range from 0 to π, and for r from 0 to 8cosθ, accordingly.
The polar integral setup will be: ∨π⁰ ∨8cosθ⁰ r drdθ. When evaluating this integral, the integral with respect to r will be straightforward, resulting in r²/2, and the limits substituted will simplify to 32cos²θ/2 = 16cos²θ. This will then be integrated with respect to θ resulting in the final solution.