Let $S$ be the set of points with polar coordinates $(r, \theta),$ where $1 \le r \le 2$ and $0 \le \theta \le \frac{\pi}{2}.$  Compute the area of $S.$
The set $S$ consists of all points in the first quadrant that lie between the circles $x^2 + y^2 = 1$ and $x^2 + y^2 = 4.$

[asy]
unitsize(1.5 cm);

filldraw(arc((0,0),1,0,90)--arc((0,0),2,90,0)--cycle,gray(0.7));
draw((-0.5,0)--(2.5,0));
draw((0,-0.5)--(0,2.5));

label("$1$", (1,0), S);
label("$2$", (2,0), S);
[/asy]

Hence, the area of $S$ is
\[\frac{4 \pi - \pi}{4} = \boxed{\frac{3 \pi}{4}}.\]