# Find the length of the curve y=(3÷4)x^(4÷3)-(3÷8)x^(2÷3)+5,1<=x<=8?

Arc length is given by:

Simplify:

Hence:

Integrate directly:

Insert the limits of integration:

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To find the length of the curve represented by the function ( y = \frac{3}{4}x^{\frac{4}{3}} - \frac{3}{8}x^{\frac{2}{3}} + 5 ) over the interval ( 1 \leq x \leq 8 ), we can use the arc length formula:

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

First, find ( \frac{dy}{dx} ):

[ \frac{dy}{dx} = \frac{d}{dx} \left(\frac{3}{4}x^{\frac{4}{3}} - \frac{3}{8}x^{\frac{2}{3}} + 5\right) ]

[ \frac{dy}{dx} = \frac{3}{4} \times \frac{4}{3}x^{\frac{1}{3}} - \frac{3}{8} \times \frac{2}{3}x^{-\frac{1}{3}} ]

[ \frac{dy}{dx} = x^{\frac{1}{3}} - \frac{1}{4}x^{-\frac{1}{3}} ]

Now, plug ( \frac{dy}{dx} ) into the arc length formula and integrate:

[ L = \int_{1}^{8} \sqrt{1 + \left(x^{\frac{1}{3}} - \frac{1}{4}x^{-\frac{1}{3}}\right)^2} , dx ]

[ L = \int_{1}^{8} \sqrt{1 + x^{\frac{2}{3}} - \frac{1}{2} + \frac{1}{16}x^{-\frac{2}{3}}} , dx ]

[ L = \int_{1}^{8} \sqrt{\frac{17}{16} + x^{\frac{2}{3}} + \frac{1}{16}x^{-\frac{2}{3}}} , dx ]

[ L = \int_{1}^{8} \sqrt{\left(\frac{\sqrt{17}}{4} + \frac{1}{4}x^{\frac{1}{3}} - \frac{1}{4}x^{-\frac{1}{3}}\right)^2} , dx ]

[ L = \int_{1}^{8} \left(\frac{\sqrt{17}}{4} + \frac{1}{4}x^{\frac{1}{3}} - \frac{1}{4}x^{-\frac{1}{3}}\right) , dx ]

Now integrate this expression over the given interval (1 \leq x \leq 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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