곡선 $\displaystyle y= \frac{1} {2} + \frac{2} {3} x^ {\frac{3} {2}} \left (1 \leq x \leq 2 \right)$ 의 길이는?

• ①$\displaystyle \frac{1} {27} \left (3^ {\frac{1} {2}} - 2^ {\frac{1} {2}} \right)$
• ②$\displaystyle \frac{1} {9} \left (3^ {\frac{1} {3}} - 2^ {\frac{1} {3}} \right)$
• ③$\displaystyle \frac{1} {9} \left (3^ {\frac{5} {2}} - 2^ {\frac{5} {2}} \right)$
• ④$\displaystyle \frac{2} {3} \left (3^ {\frac{1} {2}} - 2^ {\frac{1} {2}} \right)$
• ⑤$\displaystyle \frac{2} {3} \left (3^ {\frac{3} {2}} - 2^ {\frac{3} {2}} \right)$