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

Answer 1

The arc length is #1/2ln2+3# units.

#y=1/2lnx-1/4x^2#
#y'=1/2(1/x-x)#

Arc length is given by:

#L=int_2^4sqrt(1+1/4(1/x-x)^2)dx#

Factor out the constant and expand:

#L=1/2int_2^4sqrt(4+(1/x^2-2+x^2))dx#

Simplify:

#L=1/2int_2^4sqrt(1/x^2+2+x^2)dx#

Factorize:

#L=1/2int_2^4sqrt((1/x+x)^2)dx#

Simplify:

#L=1/2int_2^4(1/x+x)dx#

Integrate term by term:

#L=1/2[lnx+1/2x^2]_2^4#

Insert the limits of integration:

#L=1/2ln2+3#
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Answer 2

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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Answer from HIX Tutor

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.

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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