Yes -- well, 1/2 the area of a unit circle, which is why I multiply by 2. I used Sage (http://www.sagemath.org/) to create these quick results:

    f(x) = sqrt(1-x^2)
    2 * integrate(f(x),x,-1,1)

Sage replies: pi (Sage is smart enough to knows this result is equal to pi, so it doesn't provide a numerical approximation, but simply prints "pi").

    plot(f(x),x,-1,1,gridlines=True,aspect_ratio=1)

Sage replies: http://i.imgur.com/GZdnT8z.png

The function by itself produces a line equal to 1/2 the distance across a unit circle at the coordinate provided by x. Therefore a definite integral of the function on the interval -1 to 1 should produce an area equal to 1/2 that of a unit circle, or pi/2.

My Calculus tutorial: http://www.arachnoid.com/calculus/