# What is the arc length of the curve given by #y = ln(x)/2 - x^2/4 # in the interval #x in [2,4]#?

The arc length is

Arc length is given by:

Factor out the constant and expand:

Simplify:

Factorize:

Simplify:

Integrate term by term:

Insert the limits of integration:

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To find the arc length of the curve ( y = \frac{\ln(x)}{2} - \frac{x^2}{4} ) in the interval ( x \in [2,4] ), we will use the arc length formula:

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

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

[ \frac{dy}{dx} = \frac{1}{2x} - \frac{x}{2} ]

Now, we substitute ( \frac{dy}{dx} ) back into the arc length formula and integrate over the interval ( [2,4] ):

[ L = \int_2^4 \sqrt{1 + \left(\frac{1}{2x} - \frac{x}{2}\right)^2} , dx ]

After simplifying the expression inside the square root and integrating, you will get the arc length ( L ).

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