The ratio of the areas between the curves `y=cosx` and `y=cos2x` and x-axis from `x=0` to `x=pi/3` is (A) `1 : 3` (B) `2 : 1` (C) `sqrt(3) : 1` (D) none of these

Options

A. `1:2`
B. `2:1`
C. `sqrt(3):1`
D. none of these

Answer

To find the ratio of the areas between the curves \( y = \cos x \) and \( y = \cos 2x \) and the x-axis from \( x = 0 \) to \( x = \frac{\pi}{3} \), we will follow these steps:

### Step 1: Calculate the area under the curve \( y = \cos x \)
The area \( A_1 \) under the curve \( y = \cos x \) from \( x = 0 \) to \( x = \frac{\pi}{3} \) can be calculated using the definite integral:

\[ A_1 = \int_{0}^{\frac{\pi}{3}} \cos x \, dx \]

### Step 2: Evaluate the integral for \( A_1 \)
The integral of \( \cos x \) is \( \sin x \). Therefore, we evaluate:

\[ A_1 = \left[ \sin x \right]_{0}^{\frac{\pi}{3}} = \sin\left(\frac{\pi}{3}\right) - \sin(0) \]

Since \( \sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2} \) and \( \sin(0) = 0 \):

\[ A_1 = \frac{\sqrt{3}}{2} - 0 = \frac{\sqrt{3}}{2} \]

### Step 3: Calculate the area under the curve \( y = \cos 2x \)
The area \( A_2 \) under the curve \( y = \cos 2x \) from \( x = 0 \) to \( x = \frac{\pi}{3} \) is given by:

\[ A_2 = \int_{0}^{\frac{\pi}{3}} \cos 2x \, dx \]

### Step 4: Evaluate the integral for \( A_2 \)
The integral of \( \cos 2x \) is \( \frac{1}{2} \sin 2x \). Therefore, we evaluate:

\[ A_2 = \left[ \frac{1}{2} \sin 2x \right]_{0}^{\frac{\pi}{3}} = \frac{1}{2} \left( \sin\left(\frac{2\pi}{3}\right) - \sin(0) \right) \]

Since \( \sin\left(\frac{2\pi}{3}\right) = \frac{\sqrt{3}}{2} \) and \( \sin(0) = 0 \):

\[ A_2 = \frac{1}{2} \left( \frac{\sqrt{3}}{2} - 0 \right) = \frac{\sqrt{3}}{4} \]

### Step 5: Find the ratio of the areas \( A_1 \) and \( A_2 \)
Now, we can find the ratio of the areas \( A_1 \) to \( A_2 \):

\[ \text{Ratio} = \frac{A_1}{A_2} = \frac{\frac{\sqrt{3}}{2}}{\frac{\sqrt{3}}{4}} = \frac{\sqrt{3}}{2} \times \frac{4}{\sqrt{3}} = 2 \]

Thus, the ratio of the areas is \( 2:1 \).

### Final Answer
The correct option is (B) \( 2 : 1 \).