`x=arcsint , y=ln(sqrt(1-t^2)) , 0<=t<=1/2` Find the arc length of the curve on the given interval.

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Arc length of a curve C described by the parametric equations x=f(t) and y=g(t), `a<=t<=b` where f' and g' are continuous on [a,b] and C is traversed exactly once as t increases from a to b, then the length of the curve is given by,


We are given:`x=arcsin(t),y=ln(sqrt(1-t^2)), 0<=t<=1/2`







Now let's evaluate arc length by using the stated formula,







Using partial fractions integrand can be written as :


Take the constant out and use the standard integral:`int1/xdx=ln|x|+C`








Arc length of the curve on the given interval is `~~0.549`

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