Maxima 5.18.1 http://maxima.sourceforge.net
Using Lisp SBCL 1.0.30
Distributed under the GNU Public License. See the file COPYING.
Dedicated to the memory of William Schelter.
The function bug_report() provides bug reporting information.

(%i2) integrate(sqrt(r^2-x^2),x,a,r);
Is  r - a  positive, negative, or zero?

p;
               2        r        2        a               2    2
              r  asin(------)   r  asin(------) + a sqrt(r  - a )
                      abs(r)            abs(r)
(%o2)         --------------- - ---------------------------------
                     2                          2
(%i3) assume(r>0);
(%o3)                               [r > 0]
(%i4) assume(a<r);
(%o4)                               [r > a]
(%i5) integrate(sqrt(r^2-x^2),x,a,r);
                                      2    2         a   2
                          2   a sqrt(r  - a ) + asin(-) r
                     %pi r                           r
(%o5)                ------ - ----------------------------
                       4                   2
(%i6) tex(%);
$${{\pi\,r^2}\over{4}}-{{a\,\sqrt{r^2-a^2}+\arcsin \left({{a}\over{r
 }}\right)\,r^2}\over{2}}$$
(%o6)                                false
(%i7)