# What is the arc length of #f(x)=sqrt(4-x^2) # on #x in [-2,2]#?

arc length

Finally, evaluate over [-2,2]

hope that helped

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To find the arc length of ( f(x) = \sqrt{4 - x^2} ) on ( x ) in the interval ([-2, 2]), we use the formula for arc length:

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

where ( a ) and ( b ) are the endpoints of the interval, and ( \frac{dy}{dx} ) is the derivative of ( f(x) ) with respect to ( x ).

First, we find ( \frac{dy}{dx} ) by differentiating ( f(x) ) with respect to ( x ):

[ f(x) = \sqrt{4 - x^2} ] [ \frac{dy}{dx} = \frac{-x}{\sqrt{4 - x^2}} ]

Now we plug ( \frac{dy}{dx} ) into the formula for arc length:

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

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

[ = \int_{-2}^{2} \sqrt{\frac{4 - x^2 + x^2}{4 - x^2}} , dx ]

[ = \int_{-2}^{2} \sqrt{\frac{4}{4 - x^2}} , dx ]

Now, to solve this integral, we can use trigonometric substitution. Let ( x = 2\sin(\theta) ). Then, ( dx = 2\cos(\theta) , d\theta ).

When ( x = -2 ), ( \theta = -\frac{\pi}{2} ).

When ( x = 2 ), ( \theta = \frac{\pi}{2} ).

[ L = \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \sqrt{\frac{4}{4 - 4\sin^2(\theta)}} \cdot 2\cos(\theta) , d\theta ]

[ = \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \sqrt{\frac{4}{4\cos^2(\theta)}} \cdot 2\cos(\theta) , d\theta ]

[ = \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \frac{4}{|\cos(\theta)|} \cdot 2\cos(\theta) , d\theta ]

[ = \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} 8 , d\theta ]

[ = 8\theta \Bigg|_{-\frac{\pi}{2}}^{\frac{\pi}{2}} ]

[ = 8\left(\frac{\pi}{2} - \left(-\frac{\pi}{2}\right)\right) ]

[ = 8\pi ]

So, the arc length of ( f(x) = \sqrt{4 - x^2} ) on ( x ) in the interval ([-2, 2]) is ( 8\pi ).

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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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