Answer
$$25\left( {\frac{{2\pi }}{3} - \frac{{\sqrt 3 }}{2}} \right)$$
Work Step by Step
$$\eqalign{
& \int_5^{10} {\sqrt {100 - {x^2}} dx} \cr
& {\text{The integrand contains the form }}{a^2} - {x^2} \cr
& 100 - {x^2} \to a = 10 \cr
& {\text{Use the change of variable }}x = a\sin \theta \cr
& x = 10\sin \theta ,\,\,\,dx = 10\cos \theta d\theta \cr
& \cr
& {\text{Substituting}} \cr
& = \int {\sqrt {100 - 100{{\sin }^2}\theta } \left( {10\cos \theta } \right)d\theta } \cr
& = \int {10\sqrt {1 - {{\sin }^2}\theta } \left( {10\cos \theta } \right)d\theta } \cr
& = \int {10\sqrt {{{\cos }^2}\theta } \left( {10\cos \theta } \right)d\theta } \cr
& = 100\int {{{\cos }^2}\theta } d\theta \cr
& = 100\int {\left( {\frac{{1 + \cos 2\theta }}{2}} \right)} d\theta \cr
& = 50\int {\left( {1 + \cos 2\theta } \right)} d\theta \cr
& {\text{Integrating}} \cr
& = 50\theta + 25\sin 2\theta + C \cr
& = 50\theta + 50\sin \theta \cos \theta + C \cr
& {\text{Write in terms of }}x \cr
& {\text{ = }}50{\sin ^{ - 1}}\left( {\frac{x}{{10}}} \right) + 50\left( {\frac{x}{{10}}} \right)\left( {\frac{{\sqrt {100 - {x^2}} }}{{10}}} \right) + C \cr
& {\text{ = }}50{\sin ^{ - 1}}\left( {\frac{x}{{10}}} \right) + \frac{{x\sqrt {100 - {x^2}} }}{2} + C \cr
& \cr
& ,{\text{then}} \cr
& \int_5^{10} {\sqrt {100 - {x^2}} dx} = \left( {50{{\sin }^{ - 1}}\left( {\frac{x}{{10}}} \right) + \frac{{x\sqrt {100 - {x^2}} }}{2}} \right)_5^{10} \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \left( {50{{\sin }^{ - 1}}\left( {\frac{{10}}{{10}}} \right) + \frac{{10\sqrt {100 - {{10}^2}} }}{2}} \right) \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, - \left( {50{{\sin }^{ - 1}}\left( {\frac{5}{{10}}} \right) + \frac{{5\sqrt {100 - {5^2}} }}{2}} \right) \cr
& = 50\left( {\frac{\pi }{2}} \right) - 50\left( {\frac{\pi }{6}} \right) - \frac{{25\sqrt 3 }}{2} \cr
& = 25\pi - \frac{{25}}{3}\pi - \frac{{25\sqrt 3 }}{2} \cr
& = \frac{{50}}{3}\pi - \frac{{25\sqrt 3 }}{2} \cr
& = 25\left( {\frac{{2\pi }}{3} - \frac{{\sqrt 3 }}{2}} \right) \cr} $$
