Question

The area of the smaller region bounded by the circle \(x^2+y^2 = 4\) and the line \(x = 1\) is...

• A. \(\frac{4π}{3}-\sqrt{3}\)
• B. \(\frac{π}{3}-\sqrt{3}\)
• C. \(\frac{4π}{3}+\sqrt{3}\)
• D. \(\frac{π}{3}+\sqrt{3}\)

Hint:

Compute a circular segment with distance 1 from the centre.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
The circle \(x^2+y^2=4\) has radius 2. The line \(x=1\) cuts it into a small part (for \(x\ge1\)) and a large part. The smaller region is the one with \(x\) from \(1\) to \(2\).

Step 2: Set up:
By symmetry about the \(x\)-axis, the area is twice the area above the axis:
\[ A=2\int_1^2\sqrt{4-x^2}\,dx \]

Step 3: Integrate:
Using \(\int\sqrt{a^2-x^2}dx=\dfrac x2\sqrt{a^2-x^2}+\dfrac{a^2}2\sin^{-1}\dfrac xa\) with \(a=2\):
\[ \left[\frac x2\sqrt{4-x^2}+2\sin^{-1}\frac x2\right]_1^2=\pi-\left(\frac{\sqrt3}2+\frac\pi3\right)=\frac{2\pi}3-\frac{\sqrt3}2 \]

Step 4: Double:
\[ A=2\left(\frac{2\pi}3-\frac{\sqrt3}2\right)=\frac{4\pi}3-\sqrt3 \]

Step 5: Choose:
Option (A).

Final Answer:
The smaller region has area 4 pi / 3 - sqrt 3. \[ \boxed{\frac{4\pi}{3}-\sqrt3} \]