The area bounded by `f(x)={{:(sin(2x),x ge 0),(cos(2x),xlt0):}` with the x - axis, `x=-(pi)/(4) and x=(pi)/(4)` is k square units. Then, the value of 4k is equal to

Answer

To find the area bounded by the function \( f(x) \) and the x-axis from \( x = -\frac{\pi}{4} \) to \( x = \frac{\pi}{4} \), we will break it down into two parts based on the piecewise definition of \( f(x) \):

1. **For \( x < 0 \)**, \( f(x) = \cos(2x) \)
2. **For \( x \geq 0 \)**, \( f(x) = \sin(2x) \)

### Step 1: Calculate the area from \( x = -\frac{\pi}{4} \) to \( x = 0 \)

We need to find the area under the curve \( f(x) = \cos(2x) \) from \( x = -\frac{\pi}{4} \) to \( x = 0 \).

\[ \text{Area}_1 = \int_{-\frac{\pi}{4}}^{0} \cos(2x) \, dx \]

### Step 2: Evaluate the integral

The integral of \( \cos(2x) \) is:

\[ \int \cos(2x) \, dx = \frac{1}{2} \sin(2x) + C \]

Now, we evaluate the definite integral:

\[ \text{Area}_1 = \left[ \frac{1}{2} \sin(2x) \right]_{-\frac{\pi}{4}}^{0} \]

Calculating the limits:

\[ = \frac{1}{2} \sin(2 \cdot 0) - \frac{1}{2} \sin\left(2 \cdot -\frac{\pi}{4}\right) \]

\[ = \frac{1}{2} \cdot 0 - \frac{1}{2} \sin\left(-\frac{\pi}{2}\right) \]

\[ = 0 - \frac{1}{2} \cdot (-1) = \frac{1}{2} \]

### Step 3: Calculate the area from \( x = 0 \) to \( x = \frac{\pi}{4} \)

Next, we find the area under the curve \( f(x) = \sin(2x) \) from \( x = 0 \) to \( x = \frac{\pi}{4} \).

\[ \text{Area}_2 = \int_{0}^{\frac{\pi}{4}} \sin(2x) \, dx \]

### Step 4: Evaluate the integral

The integral of \( \sin(2x) \) is:

\[ \int \sin(2x) \, dx = -\frac{1}{2} \cos(2x) + C \]

Now, we evaluate the definite integral:

\[ \text{Area}_2 = \left[-\frac{1}{2} \cos(2x)\right]_{0}^{\frac{\pi}{4}} \]

Calculating the limits:

\[ = -\frac{1}{2} \cos\left(2 \cdot \frac{\pi}{4}\right) - \left(-\frac{1}{2} \cos(0)\right) \]

\[ = -\frac{1}{2} \cos\left(\frac{\pi}{2}\right) + \frac{1}{2} \cdot 1 \]

\[ = -\frac{1}{2} \cdot 0 + \frac{1}{2} = \frac{1}{2} \]

### Step 5: Total Area

Now, we add both areas to find the total area \( k \):

\[ k = \text{Area}_1 + \text{Area}_2 = \frac{1}{2} + \frac{1}{2} = 1 \]

### Step 6: Calculate \( 4k \)

Finally, we calculate \( 4k \):

\[ 4k = 4 \cdot 1 = 4 \]

### Final Answer

The value of \( 4k \) is \( \boxed{4} \).

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