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

Answer 1

#1/27[sqrt[[85^3]]-sqrt[[40^3]]]#

Given, #f[t]# = #[t^3-1, t^2-1] # i.e, #[x=t^3-1, y=t^2-1]#.....#[1]# Differentiating w.r.t. #t#,
#dx/dt=3t^2, dy/dt=2t#.....#[2]#. The arc length of a curve in parametric form is given by, #L#=#int_2^3sqrt[[dx/dt]^2+[dy/dt]^2]dt#, substituting #dx/dt and dy/dt# from ....#[2]# into the length formula #L# will give,
#L#=#int_2^3sqrt[9t^4+4t^2dt]# = #int_2^3sqrt[t^2[9t^2+4]]#=#int_2^3tsqrt[9t^2+4]dt#.
We can now make a substitution by letting #[u=9t^2+4]#........#[3]#. Differentiating .....#[3]# w.r.t #x#, #du/dt= 18t#, .i.e, #[du]/[18t]#=#dt#.
So we now have #L#=#intt sqrt[u]/[18t]dt# = #1/18intsqrt[u]dt# [ since the terms in #t#will cancel],
And after integrating w.r.t. #t# we are left with #1/18 [2/3]sqrt[u^3]# = #1/27sqrt[u^3# we now have change the bounds of integration, From ......#[3]#, #u=9t^2+4#, so when #t=2, u=40#, when #t=3, u=85#.
So length #L#=#1/27[sqrt[85^3]-sqrt[40^3]]# units.
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Answer 2

To find the arc length of the curve defined by ( f(t) = (t^3 - 1, t^2 - 1) ) on the interval ( t ) in ([2,3]), we use the formula for arc length:

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

For the given function ( f(t) ), its derivatives are:

[ \frac{dx}{dt} = 3t^2 ] [ \frac{dy}{dt} = 2t ]

Now, substitute these derivatives into the arc length formula and integrate over the interval ([2,3]):

[ L = \int_{2}^{3} \sqrt{(3t^2)^2 + (2t)^2} , dt ]

[ L = \int_{2}^{3} \sqrt{9t^4 + 4t^2} , dt ]

This integral is usually computed numerically since it doesn't have a simple closed-form solution.

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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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