The ratio of the areas bounded by `y=cosx,y=cos2x` between x=0 and `x=pi//3` and the x-axis is

Options

A. `1:2`
B. `2:1`
C. `sqrt(3):4`
D. `2sqrt(3):4-sqrt(3)`

Answer

To find the ratio of the areas bounded by the curves \( y = \cos x \) and \( y = \cos 2x \) between \( x = 0 \) and \( x = \frac{\pi}{3} \) and the x-axis, 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 \( y = \cos 2x \)

Next, we calculate the area \( A_2 \) under the curve \( y = \cos 2x \) from \( x = 0 \) to \( x = \frac{\pi}{3} \):

\[ A_2 = \int_0^{\frac{\pi}{3}} \cos 2x \, dx \]

### 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 \( \sin\left(\frac{2\pi}{3}\right) = \sin(120^\circ) = \frac{\sqrt{3}}{2} \) and \( \sin(0) = 0 \):

\[ A_2 = \frac{1}{2} \cdot \frac{\sqrt{3}}{2} = \frac{\sqrt{3}}{4} \]

### Step 5: Find the ratio of the areas

Now, we find the ratio of the areas \( A_1 \) and \( A_2 \):

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

Thus, the ratio of the areas is:

\[ \text{Ratio} = 2 : 1 \]

### Final Answer

The ratio of the areas bounded by \( y = \cos x \) and \( y = \cos 2x \) between \( x = 0 \) and \( x = \frac{\pi}{3} \) and the x-axis is \( 2 : 1 \).

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