The area bounded between curves `y^2 = x and y= |x|`

Options

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

Answer

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

### Step 1: Identify the curves
The first curve is \( y^2 = x \), which represents a rightward-opening parabola. The second curve is \( y = |x| \), which 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 set \( y^2 = x \) equal to \( y = |x| \).

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

2. For \( x < 0 \) (where \( y = -x \)):
\[ y^2 = x \implies y = -x \implies (-x)^2 = x \implies x^2 = x \implies x(x - 1) = 0 \]
This does not provide any additional points of intersection since \( x < 0 \) is not satisfied.

### Step 3: Set up the integral for the area
The area \( A \) between the curves from \( x = 0 \) to \( x = 1 \) can be found using the integral:
\[ A = \int_{0}^{1} (y_{\text{top}} - y_{\text{bottom}}) \, dx \]
Here, \( y_{\text{top}} = \sqrt{x} \) (from \( y^2 = x \)) and \( y_{\text{bottom}} = x \) (from \( y = |x| \) for \( x \geq 0 \)). Thus, we have:
\[ A = \int_{0}^{1} (\sqrt{x} - x) \, dx \]

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

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

2. Calculate \( \int_{0}^{1} x \, dx \):
\[ \int x \, dx = \frac{1}{2} x^2 \bigg|_{0}^{1} = \frac{1}{2} (1^2) - \frac{1}{2} (0^2) = \frac{1}{2} \]

Now, substituting these results back into the area formula:
\[ A = \frac{2}{3} - \frac{1}{2} \]

### Step 5: Simplify the result
To simplify \( \frac{2}{3} - \frac{1}{2} \), we find a common denominator (which is 6):
\[ A = \frac{4}{6} - \frac{3}{6} = \frac{1}{6} \]

### Final Answer
Thus, the area bounded between the curves \( y^2 = x \) and \( y = |x| \) is:
\[ \boxed{\frac{1}{6}} \text{ square units.} \]