M14-28

Official Solution:

What is the area of the smaller sector enclosed by the graphs of \(x^2 + y^2 = 4\) and \(y = |x|\)?

A. \(\frac{\pi}{4}\)
B. \(\frac{\pi}{2}\)
C. \(\pi\)
D. \(2\pi\)
E. \(3\pi\)


The equation \(x^2 + y^2 = 4\) represents a circle centered at the origin with a radius of \(\sqrt{4} = 2\).

The graph of \(y = |x|\) is shown below in blue:



The smaller sector enclosed by these graphs is the yellow sector located in the top part of the circle. Since the central angle of this sector is 90 degrees, its area is \(\frac{1}{4}\) of the circle's area (given that the circle has a total angle of 360 degrees).

The area of the circle is \(\pi r^2 = 4\pi\). Therefore, the area of the smaller sector is \(\frac{1}{4}\) of this value, which is equal to \(\pi\) square units.


Answer: C