To determine the area of the shaded region, we must calculate the area bounded by the curve \( y = x^2 \), the line \( x = 2 \), the y-axis (\( x = 0 \)), and the region between \( y = 0 \) and \( y = 4 \).
1. Geometric Analysis: Given the curve \( y = x^2 \), we can express \( x \) in terms of \( y \) as \( x = \sqrt{y} \). The line \( x = 2 \) intersects the parabola at \( y = 4 \) because \( y = (2)^2 = 4 \). Therefore, the \( y \) bounds for the integration are from 0 to 4.
2. Area Calculation with Respect to y: The shaded region is defined by the space between the parabola \( x = \sqrt{y} \) on the left and the vertical line \( x = 2 \) on the right. This means a horizontal strip within the region extends from \( x = \sqrt{y} \) to \( x = 2 \). The width of this strip is \( (2 - \sqrt{y}) \) and its height is \( dy \). The resulting area integral is:
\( A = \int_{0}^{4} (2 - \sqrt{y}) \, dy \) This expression does not match the provided options. The critical insight is to represent the shaded area using the correct integral format.
3. Comparison with Options: Option (D): \( \int_0^4 \sqrt{y} \, dy \) represents the area under the curve \( x = \sqrt{y} \), not the entire strip up to \( x = 2 \). Option (B): \( \int_0^2 \sqrt{y} \, dy \) has an incorrect upper limit; it should be 4. Options (A) and (C): \( \int_0^2 x^2 \, dx \) calculates the area beneath the parabola \( y = x^2 \) from \( x = 0 \) to \( x = 2 \), which accurately describes our shaded region!
4. Determination: The area of the shaded region is correctly formulated as: \( \int_0^2 x^2 \, dx \)
Final Selection: The correct option is (A) \( \int_0^2 x^2 \, dx \).
