Question

The line \( x = \frac{\pi}{4} \) divides the area of the region bounded by \( y = \sin x \), \( y = \cos x \) and the x-axis \( (0 \le x \le \frac{\pi}{2}) \) into areas \( A_{1} \) and \( A_{2} \). Then \( A_{1} : A_{2} \) equals

• A. 4:1
• B. 3:1
• C. 2:1
• D. 1:1

Hint:

Due to symmetry between sine and cosine around $\pi/4$, these areas are identical.

Answer 1

The Correct Option is D

Step 1: Area $A_1$
$A_{1} = \int_{0}^{\pi/4} \sin x dx = [-\cos x]_{0}^{\pi/4} = 1 - \frac{1}{\sqrt{2}} = \frac{\sqrt{2}-1}{\sqrt{2}}$.
Step 2: Area $A_2$
$A_{2} = \int_{\pi/4}^{\pi/2} \cos x dx = [\sin x]_{\pi/4}^{\pi/2} = 1 - \frac{1}{\sqrt{2}} = \frac{\sqrt{2}-1}{\sqrt{2}}$.
Step 3: Ratio
$A_{1} : A_{2} = 1 : 1$.
Final Answer: (d)