[tex]x=9\sin{t}\\dx=9\cos{t}\,dt\\ \int\limits {\sqrt{(81-x^2)}} \, dx =\int\limits {\sqrt{(81-(9\sin{t})^2}} \, 9\cos{t}\,dt\\ =\int\limits {\sqrt{(81-81\sin^2{t})}} \, 9\cos{t}\,dt =\int\limits {\sqrt{81(1-\sin^2{t})}} \, 9\cos{t}\,dt\\=\int\limits {(\sqrt{81\cos^2{t}})} \, 9\cos{t}\,dt =\int\limits {(9\cos{t})} \, 9\cos{t}\,dt =\int\limits {81\cos^2{t}}\,dt\\ =\frac{81}{2}\int\limits {(1-\cos{2t}})\,dt =\frac{81}{2}(t-\frac{\sin{2t}}{2}})+c\\ =\frac{81}{2}t-\frac{81\sin{2t}}{4}}+c[/tex]
