Maths. Section B1Application of Integrals

The smaller of the areas enclosed by the circle x2 + y2 = 4 and the l…

The smaller of the areas enclosed by the circle x² + y² = 4 and the line x + y = 2 is

Answer & explanation

Correct answer: option 2

The correct answer is Option (2) → $\pi-2$

The circle has centre O(0, 0) and radius 2. The line x + y = 2 meets it at A(2, 0) and B(0, 2), both on the axes. The smaller region is the part of the first-quadrant quarter circle that lies above the chord AB.

$Circle and line with the smaller region shaded$

So the area is the quarter circle minus the triangle OAB:

$ \displaystyle \text{Quarter circle} = \frac{1}{4}\pi(2)^2 = \pi $

$ \displaystyle \text{Triangle OAB} = \frac{1}{2}\times 2\times 2 = 2 $

$ \text{Area} = \pi - 2 $ sq. units

The same value follows by integration:

$ \displaystyle \int_{0}^{2}\left(\sqrt{4-x^2} - (2-x)\right)dx = \pi - 2 $