Answer
$$\frac{\pi }{3}$$
Work Step by Step
$$\eqalign{
& \int_3^6 {\frac{1}{{\sqrt {36 - {x^2}} }}} dx \cr
& \frac{1}{{\sqrt {36 - {x^2}} }}{\text{ has an infinite discontinuity at }}x = 6,{\text{so we can write}} \cr
& \int_3^6 {\frac{1}{{\sqrt {36 - {x^2}} }}} dx = \mathop {\lim }\limits_{b \to {6^ - }} \int_3^b {\frac{1}{{\sqrt {36 - {x^2}} }}} dx \cr
& {\text{Integrate, use }}\int {\frac{1}{{\sqrt {{a^2} - {u^2}} }}} dx = \arcsin \left( {\frac{x}{a}} \right) + C \cr
& \mathop {\lim }\limits_{b \to {6^ - }} \int_3^b {\frac{1}{{\sqrt {36 - {x^2}} }}} d\theta = \mathop {\lim }\limits_{b \to {6^ - }} \left[ {\arcsin \left( {\frac{x}{6}} \right)} \right]_3^b \cr
& = \mathop {\lim }\limits_{b \to {6^ - }} \left[ {\arcsin \left( {\frac{b}{6}} \right) - \arcsin \left( {\frac{3}{6}} \right)} \right] \cr
& = \mathop {\lim }\limits_{b \to {6^ - }} \left[ {\arcsin \left( {\frac{b}{6}} \right) - \frac{\pi }{6}} \right] \cr
& {\text{Evaluate the limit when }}b \to 6 \cr
& = \arcsin \left( {\frac{6}{6}} \right) - \frac{\pi }{6} \cr
& = \frac{\pi }{2} - \frac{\pi }{6} \cr
& = \frac{\pi }{3} \cr
& {\text{The following graph confirms the result}} \cr} $$
