The area of the region enclosed by `y=x^2 and y=sqrt(|x|)` is

Options

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

Explanation

To find the area of the region enclosed by the curves \( y = x^2 \) and \( y = \sqrt{|x|} \), we will follow these steps:

### Step 1: Identify the points of intersection
To find the area between the curves, we first need to determine where they intersect. We set the equations equal to each other:

\[ x^2 = \sqrt{|x|} \]

This equation can be split into two cases based on the definition of \(|x|\).

**Case 1:** \( x \geq 0 \)

\[ x^2 = \sqrt{x} \]

Squaring both sides gives:

\[ x^4 = x \]

Rearranging this, we get:

\[ x^4 - x = 0 \implies x(x^3 - 1) = 0 \]

Thus, \( x = 0 \) or \( x^3 = 1 \) which gives \( x = 1 \).

**Case 2:** \( x < 0 \)

\[ x^2 = \sqrt{-x} \]

Squaring both sides gives:

\[ x^4 = -x \]

Rearranging this, we have:

\[ x^4 + x = 0 \implies x(x^3 + 1) = 0 \]

Thus, \( x = 0 \) or \( x^3 = -1 \) which gives \( x = -1 \).

The points of intersection are \( x = -1 \) and \( x = 1 \).

### Step 2: Set up the integral for the area
The area \( A \) between the curves from \( x = -1 \) to \( x = 1 \) can be calculated using the integral:

\[ A = \int_{-1}^{1} (\sqrt{|x|} - x^2) \, dx \]

Since the function \( \sqrt{|x|} \) is symmetric about the y-axis, we can simplify the calculation by evaluating from \( 0 \) to \( 1 \) and then doubling the result:

\[ A = 2 \int_{0}^{1} (\sqrt{x} - x^2) \, dx \]

### Step 3: Calculate the integral
Now we compute the integral:

\[ \int_{0}^{1} (\sqrt{x} - x^2) \, dx = \int_{0}^{1} x^{1/2} \, dx - \int_{0}^{1} x^2 \, dx \]

Calculating each part:

1. For \( \int_{0}^{1} x^{1/2} \, dx \):

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

2. For \( \int_{0}^{1} x^2 \, dx \):

\[ \int x^2 \, dx = \frac{x^3}{3} \bigg|_{0}^{1} = \frac{1}{3} - 0 = \frac{1}{3} \]

Now substituting back into the integral:

\[ \int_{0}^{1} (\sqrt{x} - x^2) \, dx = \frac{2}{3} - \frac{1}{3} = \frac{1}{3} \]

### Step 4: Final area calculation
Thus, the total area is:

\[ A = 2 \cdot \frac{1}{3} = \frac{2}{3} \]

### Conclusion
The area of the region enclosed by the curves \( y = x^2 \) and \( y = \sqrt{|x|} \) is \( \frac{2}{3} \).

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