The area enclosed by the curve `|y|=sin2x,` where `x in [0,2pi].` is

Options

A. 1 sq unit
B. 2 sq unit
C. 3 sq unit
D. 4 sq unit

Answer

To find the area enclosed by the curve \( |y| = \sin(2x) \) for \( x \) in the interval \( [0, 2\pi] \), we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Curve**:
The equation \( |y| = \sin(2x) \) indicates that \( y \) can take both positive and negative values of \( \sin(2x) \). Therefore, the graph will oscillate between \( y = \sin(2x) \) and \( y = -\sin(2x) \).

2. **Determine the Points of Intersection**:
The sine function oscillates between 0 and 1. We need to find the points where \( \sin(2x) = 0 \) within the interval \( [0, 2\pi] \).
\[ 2x = n\pi \implies x = \frac{n\pi}{2} \]
For \( n = 0, 1, 2, 3, 4 \), the points are:
- \( x = 0 \)
- \( x = \frac{\pi}{2} \)
- \( x = \pi \)
- \( x = \frac{3\pi}{2} \)
- \( x = 2\pi \)

3. **Sketch the Graph**:
The graph of \( y = \sin(2x) \) will have a period of \( \pi \) and will cross the x-axis at the points identified. The graph will oscillate above and below the x-axis, creating symmetrical areas.

4. **Calculate the Area**:
The area between the curve and the x-axis from \( 0 \) to \( \frac{\pi}{2} \) can be computed. Since the graph is symmetrical, we can multiply the area from \( 0 \) to \( \frac{\pi}{2} \) by 4 to get the total area.
\[ \text{Area} = 4 \int_{0}^{\frac{\pi}{2}} \sin(2x) \, dx \]

5. **Evaluate the Integral**:
The integral \( \int \sin(2x) \, dx \) can be computed as follows:
\[ \int \sin(2x) \, dx = -\frac{1}{2} \cos(2x) + C \]
Therefore,
\[ \int_{0}^{\frac{\pi}{2}} \sin(2x) \, dx = \left[-\frac{1}{2} \cos(2x)\right]_{0}^{\frac{\pi}{2}} = -\frac{1}{2} \left( \cos(\pi) - \cos(0) \right) \]
\[ = -\frac{1}{2} \left(-1 - 1\right) = -\frac{1}{2} \times (-2) = 1 \]

6. **Final Area Calculation**:
Now, substituting back into the area formula:
\[ \text{Total Area} = 4 \times 1 = 4 \]

### Conclusion:
The area enclosed by the curve \( |y| = \sin(2x) \) for \( x \) in the interval \( [0, 2\pi] \) is \( 4 \).