Question

The value of the integral \[ \int_0^1 \int_0^{1 - y^2} y \sin (\pi(1 - x^2)^2) \, dx \, dy \] is

• A. \( \frac{1}{2 \pi} \)
• B. \( 2\pi \)
• C. \( \frac{\pi}{2} \)
• D. \( \frac{2}{\pi} \)

Hint:

When dealing with double integrals, evaluate the inner integral first, simplifying as much as possible before performing the outer integral.

Answer 1

The Correct Option is A

Step 1: Set up the integral and simplify.
The given double integral is: \[ I = \int_0^1 \int_0^{1 - y^2} y \sin (\pi(1 - x^2)^2) \, dx \, dy. \] First, evaluate the inner integral with respect to \( x \), and then integrate with respect to \( y \). Simplify the trigonometric expression and solve the integrals.

Step 2: Conclusion.
After performing the integration, the value of the integral is \( \frac{\pi}{2} \), so the correct answer is \( (C) \).