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

Use algebra to get a length of

This is the equation of a circle. Complete the square to get

The length of the upper semicircle is half the circumference.

Note I also tried to do this using the integral, but it became too complicated for me to continue when there was a much cleaner solution available.

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

[ L = \int_{a}^{b} \sqrt{1 + \left(\frac{dy}{dx}\right)^2} , dx ]

For the given curve ( y = \sqrt{x - x^2} ), differentiate it to find ( \frac{dy}{dx} ): [ \frac{dy}{dx} = \frac{1 - 2x}{2\sqrt{x - x^2}} ]

Now, square this derivative and add 1 to it: [ \left(\frac{dy}{dx}\right)^2 + 1 = \frac{(1 - 2x)^2}{4(x - x^2)} + 1 ]

Integrate this expression from the lower limit of ( x ) to the upper limit to find the arc length. Since the function is symmetric about the midpoint of its domain, we can integrate from ( x = 0 ) to ( x = \frac{1}{2} ), and then double the result to find the total length.

[ L = 2\int_{0}^{\frac{1}{2}} \sqrt{\frac{(1 - 2x)^2}{4(x - x^2)} + 1} , dx ]

Evaluate this integral to find the length of the curve.

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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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