The area between the curves `y = x^(3)` and `y = x + |x|` is equal to

Options

A. 0 sq. units
B. 2 sq. units
C. 1 sq. units
D. 3 sq. units

Answer

To find the area between the curves \( y = x^3 \) and \( y = x + |x| \), we will follow these steps:

### Step 1: Understand the curves
The curve \( y = x^3 \) is a cubic function that passes through the origin and has a point of inflection at \( x = 0 \). The curve \( y = x + |x| \) can be simplified based on the value of \( x \):
- For \( x \geq 0 \), \( |x| = x \) so \( y = x + x = 2x \).
- For \( x < 0 \), \( |x| = -x \) so \( y = x - x = 0 \).

Thus, we can rewrite the second curve as:
- \( y = 2x \) for \( x \geq 0 \)
- \( y = 0 \) for \( x < 0 \)

### Step 2: Find the points of intersection
We need to find where the curves intersect:
1. Set \( x^3 = 2x \).
2. Rearranging gives us:
\[ x^3 - 2x = 0 \]
3. Factoring out \( x \):
\[ x(x^2 - 2) = 0 \]
4. This gives us the solutions:
\[ x = 0, \quad x = \sqrt{2}, \quad x = -\sqrt{2} \]

### Step 3: Set up the integral for the area
Since we are interested in the area between the curves from \( x = 0 \) to \( x = \sqrt{2} \), we will set up the integral:
\[ \text{Area} = \int_{0}^{\sqrt{2}} (2x - x^3) \, dx \]

### Step 4: Compute the integral
1. Calculate the integral:
\[ \int (2x - x^3) \, dx = \int 2x \, dx - \int x^3 \, dx \]
\[ = x^2 - \frac{x^4}{4} + C \]
2. Evaluate the definite integral from \( 0 \) to \( \sqrt{2} \):
\[ \text{Area} = \left[ x^2 - \frac{x^4}{4} \right]_{0}^{\sqrt{2}} \]
\[ = \left( (\sqrt{2})^2 - \frac{(\sqrt{2})^4}{4} \right) - \left( 0 - 0 \right) \]
\[ = \left( 2 - \frac{4}{4} \right) \]
\[ = 2 - 1 = 1 \]

### Final Answer
The area between the curves \( y = x^3 \) and \( y = x + |x| \) is \( 1 \) square unit.

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