Question

The area of the shaded region (figure) represented by the curves \( y = x^2 \), \( 0 \leq x \leq 2 \), and the y-axis is given by:

• A. \( \int_0^2 x^2 \, dx \)
• B. \( \int_0^2 \sqrt{y} \, dy \)
• C. \( \int_0^4 x^2 \, dx \)
• D. \( \int_0^4 \sqrt{y} \, dy \)

Hint:

To find the area under a curve between two points, integrate the function with respect to the variable representing the horizontal axis.

Answer 1

The Correct Option is A

Step 1: Understanding the shaded region.
The given curves are \( y = x^2 \) and the y-axis, with \( x \) ranging from 0 to 2. The area of the shaded region is the area under the curve \( y = x^2 \) from \( x = 0 \) to \( x = 2 \). Step 2: Setting up the integral.
The area under the curve \( y = x^2 \) is given by the integral: \[ \text{Area} = \int_0^2 x^2 \, dx \] Thus, the correct answer is \( \boxed{\int_0^2 x^2 \, dx} \).