 <p>In the latest version of SymPy you will not obtain an integrated result unless you make the 'r' variable positive</p> <pre><code>&gt;&gt;&gt; var('r',positive=1) r &gt;&gt;&gt; integrate(sqrt(r**2 - x**2), (x, -r, r)) pi*r**2/2 </code></pre> <p>Notice that you get the area of a half-circle, too. Compare to</p> <pre><code>&gt;&gt;&gt; Circle((x,y),r).area pi*r**2 </code></pre>