What is the arc length of #f(x)= (3x-2)^2 # on #x in [1,3] #?

Answer 1

#L=1/6(7sqrt197-sqrt5)+1/12ln((14+sqrt197)/(2sqrt5))# units.

#f(x)=(3x−2)^2#
#f'(x)=6x−4#

Arc length is given by:

#L=int_1^3sqrt(1+(6x-4)^2)dx#
Apply the substitution #6x-4=u#:
#L=1/6int_2^14sqrt(1+u^2)du#
Apply the substitution #u=tantheta#:
#L=1/6intsec^3thetad theta#

This is a known integral:

#L=1/12[secthetatantheta+ln|sectheta+tantheta|]#

Reverse the last substitution:

#L=1/12[usqrt(1+u^2)+ln|u+sqrt(1+u^2)|]_2^14#

Hence

#L=1/6(7sqrt197-sqrt5)+1/12ln((14+sqrt197)/(2sqrt5))#
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Answer 2

To find the arc length ( L ) of the function ( f(x) = (3x - 2)^2 ) on the interval ([1, 3]), we use the formula for arc length:

[ L = \int_{a}^{b} \sqrt{1 + [f'(x)]^2} , dx ]

First, we need to find ( f'(x) ), the derivative of ( f(x) ):

[ f'(x) = 2(3x - 2)(3) = 6(3x - 2) ]

Next, we compute ([f'(x)]^2):

[ [f'(x)]^2 = [6(3x - 2)]^2 = 36(3x - 2)^2 ]

Now, we substitute this derivative into the arc length formula and evaluate the integral over the interval ([1, 3]):

[ L = \int_{1}^{3} \sqrt{1 + 36(3x - 2)^2} , dx ]

This integral represents the arc length of the function ( f(x) = (3x - 2)^2 ) on the interval ([1, 3]). The exact numerical value of this integral would require computational methods or specialized software to evaluate.

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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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