The area enclosed between the curves `y^2=x and y=|x|` is

Options

A. `1/6`
B. `1/3`
C. `2/3`
D. 1

Explanation

To find the area enclosed between the curves \( y^2 = x \) and \( y = |x| \), we will follow these steps:

### Step 1: Understand the curves
The curve \( y^2 = x \) represents a parabola that opens to the right. The equation \( y = |x| \) consists of two lines: \( y = x \) for \( x \geq 0 \) and \( y = -x \) for \( x < 0 \).

### Step 2: Find the points of intersection
To find the points where the curves intersect, we need to set \( y^2 = x \) equal to \( y = x \) and \( y = -x \).

1. For \( y = x \):
\[ x^2 = x \implies x^2 - x = 0 \implies x(x - 1) = 0 \]
This gives us \( x = 0 \) and \( x = 1 \).

2. For \( y = -x \):
\[ (-x)^2 = x \implies x^2 = x \implies x^2 - x = 0 \implies x(x - 1) = 0 \]
This gives us the same points of intersection, \( x = 0 \) and \( x = 1 \).

### Step 3: Determine the area between the curves
The area between the curves from \( x = 0 \) to \( x = 1 \) can be found by integrating the difference of the upper curve and the lower curve.

- The upper curve in this interval is \( y = x \) and the lower curve is \( y = \sqrt{x} \) (since \( y^2 = x \) implies \( y = \sqrt{x} \) for \( y \geq 0 \)).

Thus, the area \( A \) can be calculated as:
\[ A = \int_{0}^{1} (x - \sqrt{x}) \, dx \]

### Step 4: Calculate the integral
Now we compute the integral:
\[ A = \int_{0}^{1} (x - \sqrt{x}) \, dx = \int_{0}^{1} x \, dx - \int_{0}^{1} \sqrt{x} \, dx \]

Calculating each integral separately:
1. For \( \int_{0}^{1} x \, dx \):
\[ \int x \, dx = \frac{x^2}{2} \Big|_{0}^{1} = \frac{1^2}{2} - \frac{0^2}{2} = \frac{1}{2} \]

2. For \( \int_{0}^{1} \sqrt{x} \, dx \):
\[ \int \sqrt{x} \, dx = \frac{2}{3} x^{3/2} \Big|_{0}^{1} = \frac{2}{3} (1^{3/2} - 0^{3/2}) = \frac{2}{3} \]

Now substituting back into the area calculation:
\[ A = \frac{1}{2} - \frac{2}{3} \]

### Step 5: Simplify the result
To combine the fractions, we find a common denominator:
\[ A = \frac{3}{6} - \frac{4}{6} = \frac{-1}{6} \]

However, since we are calculating area, we take the absolute value:
\[ A = \frac{1}{6} \text{ square units} \]

### Final Answer
The area enclosed between the curves \( y^2 = x \) and \( y = |x| \) is \( \frac{1}{6} \) square units.

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