Question

Let \[ S = \{(x, y) \in \mathbb{R}^2 : x \geq 0, \sqrt{4 - (x - 2)^2} \leq y \leq \sqrt{9 - (x - 3)^2} \}. \] Then the area of \( S \) equals .............

Hint:

For regions between curves, calculate the area of each region and subtract to find the total area.

Answer 1

Correct Answer: 7.8 - 7.9

Step 1: Identify the curves. 
The region \( S \) is bounded between two curves: \[ y = \sqrt{4 - (x - 2)^2} \text{and} y = \sqrt{9 - (x - 3)^2}, \] which are portions of circles. The first curve is a semicircle with radius 2, centered at \( (2, 0) \), and the second curve is a semicircle with radius 3, centered at \( (3, 0) \).

Step 2: Sketch and understand the region. 
The region \( S \) represents the area between these two semicircles. The area of \( S \) can be computed by finding the area of the larger semicircle and subtracting the area of the smaller semicircle.

Step 3: Calculate the area. 
The area of a full circle is \( \pi r^2 \), so the area of a semicircle is half that: - The area of the larger semicircle is \( \frac{1}{2} \pi (3)^2 = \frac{9\pi}{2} \). - The area of the smaller semicircle is \( \frac{1}{2} \pi (2)^2 = 2\pi \). Thus, the area of \( S \) is: \[ \text{Area of } S = \frac{9\pi}{2} - 2\pi = \frac{5\pi}{2}. \]

Step 4: Conclusion. 
The area of \( S \) is \( 7.85\).