The area (in sq. units) bounded by `y=x|x|` and the line `y=x` is equal to

Options

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

Answer

To find the area bounded by the curves \( y = x|x| \) and the line \( y = x \), we will follow these steps:

### Step 1: Understand the function \( y = x|x| \)
The function \( y = x|x| \) can be rewritten as:
- \( y = x^2 \) for \( x \geq 0 \)
- \( y = -x^2 \) for \( x < 0 \)

### Step 2: Find the points of intersection
We need to find the points where the curves intersect:
1. For \( x \geq 0 \):
\[ x^2 = x \implies x^2 - x = 0 \implies x(x - 1) = 0 \]
This gives us \( x = 0 \) and \( x = 1 \).

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

Thus, the points of intersection are \( (0, 0) \), \( (1, 1) \), and \( (-1, -1) \).

### Step 3: Sketch the curves
- The curve \( y = x^2 \) is a parabola opening upwards.
- The curve \( y = -x^2 \) is a parabola opening downwards.
- The line \( y = x \) is a diagonal line passing through the origin.

### Step 4: Set up the integrals for area calculation
The area between the curves can be calculated by integrating the difference of the functions over the intervals defined by the intersection points.

1. For \( x \) from \( 0 \) to \( 1 \):
\[ \text{Area}_1 = \int_{0}^{1} (x - x^2) \, dx \]

2. For \( x \) from \( -1 \) to \( 0 \):
\[ \text{Area}_2 = \int_{-1}^{0} (x - (-x^2)) \, dx = \int_{-1}^{0} (x + x^2) \, dx \]

### Step 5: Calculate the integrals
1. Calculate \( \text{Area}_1 \):
\[ \text{Area}_1 = \int_{0}^{1} (x - x^2) \, dx = \left[ \frac{x^2}{2} - \frac{x^3}{3} \right]_{0}^{1} = \left( \frac{1}{2} - \frac{1}{3} \right) = \frac{3}{6} - \frac{2}{6} = \frac{1}{6} \]

2. Calculate \( \text{Area}_2 \):
\[ \text{Area}_2 = \int_{-1}^{0} (x + x^2) \, dx = \left[ \frac{x^2}{2} + \frac{x^3}{3} \right]_{-1}^{0} = \left( 0 - \left( \frac{1}{2} - \frac{1}{3} \right) \right) = -\left( \frac{3}{6} - \frac{2}{6} \right) = -\frac{1}{6} \]

### Step 6: Total area
The total area is the sum of the two areas:
\[ \text{Total Area} = \text{Area}_1 + \text{Area}_2 = \frac{1}{6} + \frac{1}{6} = \frac{2}{6} = \frac{1}{3} \]

### Final Answer
The area bounded by \( y = x|x| \) and the line \( y = x \) is \( \frac{1}{3} \) square units.