The area between the arms of the curve `|y|=x^(3)` from `x = 0` to `x = 2` is

Options

A. 2
B. 4
C. 8
D. 16

Answer

To find the area between the arms of the curve \(|y| = x^3\) from \(x = 0\) to \(x = 2\), we can follow these steps:

### Step 1: Understand the Curve
The equation \(|y| = x^3\) implies two equations:
1. \(y = x^3\) (for the upper arm)
2. \(y = -x^3\) (for the lower arm)

### Step 2: Identify the Area to be Calculated
We need to find the area between the curves \(y = x^3\) and \(y = -x^3\) from \(x = 0\) to \(x = 2\). The area between these two curves can be calculated by integrating the difference between the upper curve and the lower curve.

### Step 3: Set Up the Integral
The area \(A\) can be expressed as:
\[ A = \int_{0}^{2} (x^3 - (-x^3)) \, dx \]
This simplifies to:
\[ A = \int_{0}^{2} (x^3 + x^3) \, dx = \int_{0}^{2} 2x^3 \, dx \]

### Step 4: Calculate the Integral
Now, we can calculate the integral:
\[ A = 2 \int_{0}^{2} x^3 \, dx \]
The integral of \(x^3\) is:
\[ \int x^3 \, dx = \frac{x^4}{4} \]
Thus, we evaluate:
\[ A = 2 \left[ \frac{x^4}{4} \right]_{0}^{2} \]
Calculating the definite integral:
\[ A = 2 \left( \frac{2^4}{4} - \frac{0^4}{4} \right) = 2 \left( \frac{16}{4} - 0 \right) = 2 \times 4 = 8 \]

### Final Answer
The area between the arms of the curve \(|y| = x^3\) from \(x = 0\) to \(x = 2\) is \(8\) square units.

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