Question

In the given figure, area of the shaded portion is equal to

• A. 3
• B. \(\frac{3\pi}{2}\)
• C. \(3\left(\frac{\pi}{2}-1\right)\)
• D. \(3(\pi-1)\)

Hint:

Shaded area \(=\int_0^{\pi/2}(3-3\cos x)\,dx\).

Answer 1

The Correct Option is C

Step 1: Understanding the Figure:
The shaded region lies between the line \(y=3\) on top and the curve \(y=3\cos x\) below. It runs from \(x=0\) to the line \(x=\frac{\pi}{2}\).

Step 2: Set up the area:
Area between two curves is the integral of (upper minus lower): \[ A=\int_0^{\pi/2}\left(3-3\cos x\right)dx \]

Step 3: Integrate:
\[ A=\left[3x-3\sin x\right]_0^{\pi/2}=\left(\frac{3\pi}{2}-3\right)-0 \]

Step 4: Simplify:
\[ A=\frac{3\pi}{2}-3=3\left(\frac{\pi}{2}-1\right) \]

Step 5: Check the options:
Option 1 is 3, which is the area under the curve, the unshaded part. Option 2 is \(\frac{3\pi}{2}\), the whole rectangle of height 3 and width \(\frac{\pi}{2}\), which also includes the unshaded part. Option 4 uses \(\pi\) instead of \(\frac{\pi}{2}\). So option 3 is right.

Final Answer:
The area is \(3\left(\frac{\pi}{2}-1\right)\) square units, option 3. \[ \boxed{3\left(\frac{\pi}{2}-1\right)} \]