What is the arc length of the curve given by #r(t)= (9sqrt(2),e^(9t),e^(-9t))# on # t in [3,4]#?

Answer 1

A zeroth-order approximation gives #2sinh(36)-2sinh(27)# units.

#r(t)=(9sqrt2, e^(9t), e^(−9t))#
#r'(t)=(0, 9e^(9t), -9e^(−9t))#

Arc length is given by:

#L=int_3^4sqrt(0+81e^(18t)+81e^(-18t))dt#

Simplify:

#L=9int_3^4sqrt(e^(18t)+e^(-18t))dt#
Apply the identity #coshx=1/2(e^x+e^-x)#:
#L=9int_3^4sqrt(2cosh(18t))dt#
Apply the identity #cosh2x=2cosh^2x-1#:
#L=9sqrt2int_3^4sqrt(2cosh^2(9t)-1)dt#

Factor out the larger piece:

#L=18int_3^4cosh(9t)sqrt(1-1/2sech^2(9t))dt#
For #t in [3, 4]#, #1/2sech^2(9t)<1#. Take the series expansion of the square root:
#L=18int_3^4cosh(9t){sum_(n=0)^oo((1/2),(n))(-1/2sech^2(9t))^n}dt#
Isolate the #n=0# term:
#L=18int_3^4cosh(9t)dt+18sum_(n=1)^oo((1/2),(n))(-1/2)^nint_3^4sech^(2n-1)(9t)dt#

A zeroth-order approximation gives:

#L~~2[sinh(9t)]_ 3^4#

Hence:

#L~~2sinh(36)-2sinh(27)#
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Answer 2

To find the arc length of the curve given by ( r(t) = (9\sqrt{2}, e^{9t}, e^{-9t}) ) on ( t ) in the interval ([3,4]), we use the formula for arc length of a curve defined by vector-valued functions:

[ s = \int_{a}^{b} |r'(t)| , dt ]

where ( r(t) ) is the vector-valued function representing the curve, and ( |r'(t)| ) is the magnitude of its derivative.

In this case, the derivative ( r'(t) ) is:

[ r'(t) = \left(0, 9e^{9t}, -9e^{-9t}\right) ]

The magnitude of ( r'(t) ) is:

[ |r'(t)| = \sqrt{0^2 + (9e^{9t})^2 + (-9e^{-9t})^2} = \sqrt{81e^{18t} + 81e^{-18t}} ]

The integral for arc length becomes:

[ s = \int_{3}^{4} \sqrt{81e^{18t} + 81e^{-18t}} , dt ]

You can evaluate this integral to find the arc length of the curve over the given interval.

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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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