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

Answer 1

#"Arclength " approx 10.601267#

#f(t)=(t^2sqrt(t-1),t^2+t-1)#
Formula for parametric arclength: #color(blue)(L = int_a^b sqrt((dx/dt)^2+(dy/dt)^2)" "dt)#
First, find #f'(t)# by differentiating #f(t)#
#dx/dt=(t^2)(1/2(t-1)^(-1/2))+(2t)(t-1)^(1/2)#
#dx/dt=frac{t^2}{2sqrt(t-1)}+2tsqrt(t-1)=frac{t^2+4t(t-1)}{2sqrt(t-1)}#
#dy/dt=2t+1#
Plug in expressions for #dx/dt# and #dy/dt# into the arclength formula: #L=int_2^3sqrt((frac{t^2+4t(t-1)}{2sqrt(t-1)})^2+(2t+1)^2)" "dt#
Use a calculator to evaluate: #"Arclength " approx 10.601267#
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Answer 2

To find the arc length of the curve represented by ( f(t) = (t^2\sqrt{t-1}, t^2+t-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 ]

Where ( a ) and ( b ) are the lower and upper bounds of the interval respectively. In this case, ( a = 2 ) and ( b = 3 ).

We differentiate ( x ) and ( y ) with respect to ( t ) to get:

[ \frac{dx}{dt} = \frac{d}{dt} (t^2\sqrt{t-1}) = 2t\sqrt{t-1} + \frac{t^2}{2\sqrt{t-1}} ]

[ \frac{dy}{dt} = \frac{d}{dt} (t^2+t-1) = 2t+1 ]

Substitute these into the arc length formula and integrate from 2 to 3:

[ L = \int_{2}^{3} \sqrt{\left(2t\sqrt{t-1} + \frac{t^2}{2\sqrt{t-1}}\right)^2 + (2t+1)^2} , dt ]

Evaluate this integral to find the arc length of the curve on the interval ([2,3]).

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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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