Question

Find the area of the circle \((x-2)^2+y^2=4\).

Hint:

This is a circle of radius 2; area = π×2² (or verify via ∫√(4−X²)dX).

Answer 1

Step 1: Using the direct geometry formula instead:
For any circle of radius \(r\), the enclosed area is the standard formula \(\pi r^2\) — this bypasses integration entirely, and serves as a quick cross-check of the integral-based method.

Step 2: Reading the radius from the equation:
Comparing \((x-2)^2+y^2=4\) with \((x-h)^2+(y-k)^2=r^2\) gives centre \((2,0)\) and \(r^2=4\Rightarrow r=2\).

Step 3: Applying the formula:
Area \(=\pi r^2=\pi(2)^2=4\pi\), matching the integration-based result exactly.

Final Answer:
\[ \boxed{4\pi \text{ sq. units}} \]