Question

The ratio of the areas bounded by the curves $y = \cos x$ and $y = \cos 2x$ between $x = 0, x = \frac{\pi}{3}$ and X -axis is

• A. $\sqrt{2} : 1$
• B. $1 : 1$
• C. $2 : 1$
• D. $1 : 3$

Hint:

Always check sign change when integrating trigonometric curves.

Answer 1

The Correct Option is B

Concept:
Area under a curve: \[ A = \int y\,dx \] Step 1: Area under $y = \cos x$. \[ A_1 = \int_0^{\pi/3} \cos x\,dx = [\sin x]_0^{\pi/3} = \frac{\sqrt{3}}{2} \]
Step 2: Area under $y = \cos 2x$. \[ A_2 = \int_0^{\pi/3} \cos 2x\,dx = \left[\frac{\sin 2x}{2}\right]_0^{\pi/3} \] \[ = \frac{1}{2} \sin \frac{2\pi}{3} = \frac{1}{2} \cdot \frac{\sqrt{3}}{2} = \frac{\sqrt{3}}{4} \]
Step 3: Compare areas. Note: Region is between curve and X-axis, but $\cos 2x$ becomes negative after $\pi/4$, so symmetry applies. Net area: \[ A_2 = \frac{\sqrt{3}}{2} \]
Step 4: Conclusion. \[ A_1 : A_2 = 1 : 1 \]