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Q5) -1∫1 ​ ₀∫ √1-x² 1/√x²+y² dydx= a) 3π/2 b) 2π (C) π d) π/2

User Aidenhjj
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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.

User Rup
by
7.7k points
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