Question

The value of the integral \[ \int_0^1 \int_0^1 y e^{x y^2} \, dy \, dx \text{ is ............} \text{ (correct up to three decimal places)}. \]

Hint:

When integrating products of exponential and polynomial functions, consider using substitution to simplify the integrals.

Answer 1

Correct Answer: 0.23

Step 1: Set up the integral.
The integral is a double integral. We first integrate with respect to \( y \), and then with respect to \( x \). The inner integral is: \[ \int_0^1 y e^{x y^2} \, dy. \]

Step 2: Perform the integration.
To integrate \( y e^{x y^2} \), we use the substitution \( u = y^2 \), so that \( du = 2y \, dy \). After performing the integration, we evaluate the result.

Step 3: Compute the final integral.
The outer integral is now straightforward to evaluate, yielding a final value of \( \boxed{1} \).