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

The correct answer is Option (3) → \( 3\left(\displaystyle\frac{\pi}{2}-1\right) \)

The shaded region lies between the line y = 3 (above) and the curve y = 3 cos x (below), from x = 0 to \( x=\frac{\pi}{2} \).

    \( \displaystyle \text{Area}=\int_0^{\pi/2}(3-3\cos x)\,dx \)

    \( \displaystyle =\left[3x-3\sin x\right]_0^{\pi/2} \)

    \( \displaystyle =\left(\frac{3\pi}{2}-3\right)-0 \)

    \( \displaystyle =3\left(\frac{\pi}{2}-1\right) \)

Shortcut: Area of the rectangle of sides \( \frac{\pi}{2} \) and 3 minus the area under the curve:

    \( \displaystyle \frac{3\pi}{2}-\int_0^{\pi/2}3\cos x\,dx=\frac{3\pi}{2}-3 \)

    \( \displaystyle =3\left(\frac{\pi}{2}-1\right) \)

Why the other options are incorrect

• Option (1) 3 — Incorrect. 3 is the area under the curve \( y=3\cos x \), the unshaded part of the rectangle.

• Option (2) \( \displaystyle\frac{3\pi}{2} \) — Incorrect. \( \frac{3\pi}{2} \) is the area of the whole rectangle, including the unshaded part below the curve.

• Option (4) \( 3(\pi-1) \) — Incorrect. \( 3(\pi-1) \) takes the width of the rectangle as π instead of \( \frac{\pi}{2} \).