Which of the following is the possible value/values of c for which the area of the figure bounded by the curves `y=sin 2x`, the straight lines `x=pi//6, x=c` and the abscissa axis is equal to 1/2?

Options

A. `-(pi)/(6)`
B. `(pi)/(3)`
C. `(pi)/(6)`
D. none of these

Explanation

To find the possible values of \( c \) for which the area bounded by the curve \( y = \sin(2x) \), the lines \( x = \frac{\pi}{6} \), \( x = c \), and the x-axis is equal to \( \frac{1}{2} \), we can follow these steps:

### Step 1: Calculate the area from \( x = \frac{\pi}{6} \) to \( x = c \)

The area \( A \) under the curve from \( x = \frac{\pi}{6} \) to \( x = c \) can be expressed as:
\[ A = \int_{\frac{\pi}{6}}^{c} \sin(2x) \, dx \]
We want this area to equal \( \frac{1}{2} \).

### Step 2: Evaluate the integral

To evaluate the integral, we find the antiderivative of \( \sin(2x) \):
\[ \int \sin(2x) \, dx = -\frac{1}{2} \cos(2x) + C \]
Thus, we can write:
\[ A = \left[-\frac{1}{2} \cos(2x)\right]_{\frac{\pi}{6}}^{c} \]

### Step 3: Substitute the limits into the integral

Substituting the limits into the integral gives:
\[ A = -\frac{1}{2} \cos(2c) + \frac{1}{2} \cos\left(\frac{\pi}{3}\right) \]
Since \( \cos\left(\frac{\pi}{3}\right) = \frac{1}{2} \), we have:
\[ A = -\frac{1}{2} \cos(2c) + \frac{1}{4} \]

### Step 4: Set the area equal to \( \frac{1}{2} \)

We set the area equal to \( \frac{1}{2} \):
\[ -\frac{1}{2} \cos(2c) + \frac{1}{4} = \frac{1}{2} \]

### Step 5: Solve for \( \cos(2c) \)

Rearranging the equation gives:
\[ -\frac{1}{2} \cos(2c) = \frac{1}{2} - \frac{1}{4} \]
\[ -\frac{1}{2} \cos(2c) = \frac{1}{4} \]
Multiplying both sides by -2:
\[ \cos(2c) = -\frac{1}{2} \]

### Step 6: Find the values of \( c \)

The general solutions for \( \cos(2c) = -\frac{1}{2} \) are:
\[ 2c = \frac{2\pi}{3} + 2k\pi \quad \text{or} \quad 2c = \frac{4\pi}{3} + 2k\pi, \quad k \in \mathbb{Z} \]
Thus, dividing by 2 gives:
\[ c = \frac{\pi}{3} + k\pi \quad \text{or} \quad c = \frac{2\pi}{3} + k\pi \]

### Step 7: Find specific values for \( c \)

For \( k = 0 \):
- \( c = \frac{\pi}{3} \)
- \( c = \frac{2\pi}{3} \)

For \( k = -1 \):
- \( c = -\frac{2\pi}{3} \)
- \( c = -\frac{\pi}{3} \)

Thus, the possible values of \( c \) are:
\[ c = -\frac{\pi}{3}, \frac{\pi}{3}, -\frac{2\pi}{3}, \frac{2\pi}{3} \]