Question

The area of the region bounded by the curve $y^2 = x$ between $x = 0$ and $x = 1$ is:

• A. $3$ sq units
• B. $2$ sq units
• C. $4$ sq units
• D. $3 \frac{1}{2}$ sq units

Hint:

When calculating areas under curves, always integrate with respect to the variable that represents the width of the region.

Answer 1

The Correct Option is B

The given curve is $y^2 = x$, which represents a parabola. We need to find the area enclosed by the curve between $x = 0$ and $x = 1$. First, solve for $y$: \[ y = \sqrt{x}. \] To find the area, we integrate the function from $x = 0$ to $x = 1$: \[ \text{Area} = \int_0^1 \sqrt{x} \, dx. \] The integral of $\sqrt{x}$ is: \[ \int \sqrt{x} \, dx = \frac{2}{3} x^{3/2}. \] Evaluating this from 0 to 1: \[ \text{Area} = \left[\frac{2}{3} x^{3/2} \right]_0^1 = \frac{2}{3} (1) - 0 = \frac{2}{3}. \] Thus, the area is 2 square units.