Question

If \[ \int_0^1 \int_0^{\sqrt{1 - (y - 1)^2}} f(x, y) \, dx \, dy \] equals \[ \int_0^1 \int_0^x f(x, y) \, dy \, dx, \] then \( \alpha(x) \) and \( \beta(x) \) are

• A. \( \alpha(x) = x, \beta(x) = 1 + \sqrt{1 - (x - 2)^2} \)
• B. \( \alpha(x) = x, \beta(x) = 1 - \sqrt{1 - (x - 2)^2} \)
• C. \( \alpha(x) = 1 + \sqrt{1 - (x - 2)^2}, \beta(x) = x \)
• D. \( \alpha(x) = 1 + \sqrt{1 - (x - 2)^2}, \beta(x) = 1 - \sqrt{1 - (x - 2)^2} \)

Hint:

When solving integral equations, carefully manipulate the bounds and integrands to identify relationships between the functions involved.

Answer 1

The Correct Option is B

Step 1: Set up the integration.
The integral equation represents a relationship between two functions \( \alpha(x) \) and \( \beta(x) \). We need to compare the two integrals and find the corresponding functions.
Step 2: Solve for \( \alpha(x) \) and \( \beta(x) \).
By solving the integrals and comparing both sides, we find that \( \alpha(x) = x \) and \( \beta(x) = 1 - \sqrt{1 - (x - 2)^2} \).
Step 3: Conclusion.
Thus, the correct answer is (B).