곡선의 길이

$\displaystyle f ( x ) = \frac{1}{3} ( x ^{2} + 2 ) ^{\frac{3}{2}}$라 하면 구하는 길이는 $\displaystyle \int _{0} ^{6} {\sqrt{1 + \left\{ f' ( x ) \right.} ^{2}} dx$$\displaystyle = \int _{0} ^{6} {\sqrt{1 + \left\{ \frac{1}{2} ( x ^{2} + 2 ) ^{\frac{1}{2}} 2 x \right.} ^{2}} dx$ $\displaystyle = \int _{0} ^{6} {\sqrt{1 + \left\{ x ^{2} ( x ^{2} + 2 ) \right.}} dx$$\displaystyle = \int _{0} ^{6} {\sqrt{( x ^{2} + 1 ) ^{2}}} dx$$\displaystyle = \int _{0} ^{6} ( x ^{2} + 1 ) dx = [ \frac{1}{3} x ^{3} + x ] _{0} ^{6}$$= 72 + 6 = 78$