which integral gives the length of the graph of y= arcsin(x/2) between x=a and x=b, where 0<a<b<pie/2?

Answer 1

Since the derivative of arcsin(x/2) is (1/2)/sqrt[ 1 - (x/2)² ] ,

after some algebra, the desired integral is:

∫ dx sqrt [ (5 - x² ) / ( 4 - x² ) ] limits a to b

Answer 2

y' = 1/(1 - x²/4)^1/2·1/2 = 1(4 - x²)^1/2

Length of curve s = ∫_a^b(1 + 1/(4 - x²))^1/2dx