Question

Area bounded by \( y=\sin^2 x \), \( x=\frac{\pi}{2} \), \( x=\pi \)

• A. \( \frac{\pi}{2} \)
• B. \( \frac{\pi}{4} \)
• C. \( \frac{\pi}{8} \)
• D. \( \frac{\pi}{16} \)
• E. \( 2\pi \)

Hint:

Always convert powers of trig into identities.

Answer 1

The Correct Option is B

Concept: Use identity \( \sin^2 x = \frac{1-\cos2x}{2} \)

Step 1: Write integral: \[ \int_{\pi/2}^{\pi} \sin^2 x dx \]

Step 2: Substitute identity: \[ = \int \frac{1-\cos2x}{2} dx \]

Step 3: Split: \[ = \frac{1}{2}\int 1 dx - \frac{1}{2}\int \cos2x dx \]

Step 4: Integrate: \[ = \frac{x}{2} - \frac{\sin2x}{4} \]

Step 5: Apply limits: \[ = \frac{\pi}{4} \]