What is the arc length of #r(t)=(t^2,2t,4-t)# on #tin [0,5]#?

Answer 1

#L=5sqrt105+5/2ln(2sqrt5+sqrt21)#

#r(t)=(t^2,2t,4−t)#
#r'(t)=(2t,2,−1)#

Arc length is given by:

#L=int_0^5sqrt(4t^2+4+1)dt#

Simplify:

#L=int_0^5sqrt(4t^2+5)dt#
Apply the substitution #2t=sqrt5tantheta#:
#L=5/2intsec^3thetad theta#

This is a known integral:

#L=5/2[secthetatantheta+ln|sectheta+tantheta|]#

Reverse the substitution:

#L=[tsqrt(4t^2+5)+5/2ln|2t+sqrt(4t^2+5)|]_0^5#

Hence

#L=5sqrt105+5/2ln(2sqrt5+sqrt21)#
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Answer 2

To find the arc length of the curve defined by ( r(t) = (t^2, 2t, 4 - t) ) on the interval ( [0, 5] ), we first need to calculate the derivative of ( r(t) ) with respect to ( t ), which gives us the velocity vector:

[ r'(t) = (2t, 2, -1) ]

Then, we compute the magnitude of this velocity vector:

[ | r'(t) | = \sqrt{(2t)^2 + 2^2 + (-1)^2} = \sqrt{4t^2 + 4 + 1} = \sqrt{4t^2 + 5} ]

Now, we integrate this magnitude over the interval ( [0, 5] ) to find the arc length:

[ \text{Arc Length} = \int_{0}^{5} | r'(t) | , dt = \int_{0}^{5} \sqrt{4t^2 + 5} , dt ]

This integral may require techniques such as trigonometric substitution or integration by parts to evaluate, depending on the preferred method of integration. Once evaluated, it will give us the arc length of the curve on the interval ( [0, 5] ).

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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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