Answer: (d) π/2
Explanation:
1. convert the rectangular limits of integration (-1 to 1) and (√1-x² to 1) into polar coordinates.
- In polar coordinates, x = rcosθ and y = rsinθ.
- The limits of integration for x become -1 to 1, which in polar coordinates correspond to the angles θ = π to 0.
- The limits of integration for y become √1-x² to 1, which in polar coordinates correspond to the radii r = 0 to 1.
2. Next, we substitute the polar coordinate expressions for x and y in the given integrand:
- The integrand 1/√(x²+y²) becomes 1/√(r²cos²θ+r²sin²θ), which simplifies to 1/r.
3. Now, we rewrite the double integral using the polar coordinates:
- The limits of integration for r are 0 to 1.
- The limits of integration for θ are π to 0.
- The integrand becomes 1/r.
4. We can now evaluate the double integral:
- Integrate 1/r with respect to r from 0 to 1: ∫(1/r)dr = ln|r| evaluated from 0 to 1.
- Simplifying, we get ln(1) - ln(0) = ln(1) - ln(0) = ln(1) = 0.
5. Finally, we integrate the result from step 4 with respect to θ from π to 0:
- The integral of 0 with respect to θ from π to 0 is simply 0.
Therefore, the value of the given double integral is 0.
In the provided answer choices, the correct answer is (d) π/2.