Question

The area of the region
\(R=\left\{(x,y) \in \R^2\ : 0\le x \le1,0 \le y \le 1\ \text{and}\ \frac{1}{4} \le xy \le \frac{1}{2} \right\}\)
is __________ (rounded off to two decimal places).

Answer 1

Correct Answer: 0.24

The correct answer is 0.24 to 0.26 (approx).

To calculate the area of the region \( R \), we need to solve for the limits of integration that satisfy the inequality conditions \( \frac{1}{4} \leq xy \leq \frac{1}{2} \). Solving for \( y \) in terms of \( x \), we get the bounds of integration for \( y \) given \( x \), and integrate over the interval \( 0 \leq x \leq 1 \). The area is computed using a double integral: \[ \int_{0}^{1} \int_{\frac{1}{4x}}^{\frac{1}{2x}} dy \, dx \] After evaluating this integral, the area is found to be approximately 0.24.