# How do you find the length of the curve #y=sqrt(x-x^2)+arcsin(sqrt(x))#?

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To find the length of the curve given by ( y = \sqrt{x - x^2} + \arcsin(\sqrt{x}) ), you can use the arc length formula for a curve ( y = f(x) ) from ( x = a ) to ( x = b ):

[ L = \int_{a}^{b} \sqrt{1 + (f'(x))^2} , dx ]

First, find the derivative of ( y ) with respect to ( x ) ( ( y' ) ):

[ y' = \frac{1 - 2x}{2\sqrt{x - x^2}} + \frac{1}{2\sqrt{x(1 - x)}} ]

Now, plug ( y' ) into the arc length formula and integrate from the starting point (( x = a )) to the ending point (( x = b )). Since the curve equation involves square roots and arcsine, the integration may not have a simple closed-form solution and might require numerical methods or special techniques.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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