# How do you find the arc length of the curve # y = (3/2)x^(2/3)# from [1,8]?

Therefore:

What normally what we would do is try to find a way to complete the square. Let's just make this look nicer for now.

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To find the arc length of the curve ( y = \frac{3}{2}x^{\frac{2}{3}} ) from ( x = 1 ) to ( x = 8 ), we use the formula for arc length:

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

First, we need to find ( \frac{dy}{dx} ) which represents the derivative of ( y ) with respect to ( x ):

[ \frac{dy}{dx} = \frac{d}{dx}\left(\frac{3}{2}x^{\frac{2}{3}}\right) ] [ = \frac{3}{2} \cdot \frac{2}{3}x^{\frac{2}{3} - 1} ] [ = x^{-\frac{1}{3}} ]

Now, we can substitute ( \frac{dy}{dx} ) into the arc length formula:

[ L = \int_{1}^{8} \sqrt{1 + \left(x^{-\frac{1}{3}}\right)^2} , dx ]

[ L = \int_{1}^{8} \sqrt{1 + x^{-\frac{2}{3}}} , dx ]

This integral can be solved using various techniques, such as substitution or integration by parts. Once integrated, you will have the arc length of the curve over the given interval [1, 8].

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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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