Question

The area in square units of the region bounded by the curve \(y = \sqrt{16-x^2}\) and lines \(x = 0,x = 4\) above the X-axis is

• A. \(16π\)
• B. \(12π\)
• C. \(8π\)
• D. \(4π\)

Hint:

The curve y = root of (16 - x squared) is the upper half of a circle of radius 4.

Answer 1

The Correct Option is D

Step 1: Approach
Use geometry instead of calculus.

Step 2: Shape
Squaring $y=\sqrt{16-x^2}$ gives $x^2+y^2=16$, a circle of radius 4. With $y\ge0$ and $0\le x\le4$ the region is exactly one quarter of the circle.

Step 3: Area
\[ \frac14\pi(4)^2=4\pi \]
Option (D).

Final Answer:
The region is a quarter of a circle of radius 4, so its area is 4 pi, option (D). \[ \boxed{4\pi} \]