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

Answer 1

#L=1/10(9sqrt130-4sqrt26)+1/(10sqrt10)ln((18+5sqrt13)/(8+sqrt65))# units.

#f(t)=(t-3t^2,t^2-t)#
#f'(t)=(1-6t,2t-1)#

Arclength is given by:

#L=int_1^2sqrt((1-6t)^2+(2t-1)^2)dt#

Expand the squares and combine terms:

#L=int_1^2sqrt(40t^2-16t+2)dt#

Complete the square in the square root:

#L=sqrt(2/5)int_1^2sqrt((10t-2)^2+1)dt#
Apply the substitution #10t-2=tantheta#:
#L=1/(5sqrt10)intsec^3thetad theta#

This is a known integral:

#L=1/(10sqrt10)[secthetatantheta+ln|sectheta+tantheta|]#

Reverse the substitution:

#L=1/(10sqrt10)[(10t-2)sqrt((10t-2)^2+1)+ln|(10t-2)+sqrt((10t-2)^2+1)|]_1^2#

Hence

#L=1/(10sqrt10){90sqrt13-8sqrt65+ln((18+5sqrt13)/(8+sqrt65))}#

Simplify:

#L=1/10(9sqrt130-4sqrt26)+1/(10sqrt10)ln((18+5sqrt13)/(8+sqrt65))#
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Answer 2

To find the arc length of the curve ( (t - 3t^2, t^2 - t) ) on the interval ( t ) from 1 to 2, we use the arc length formula:

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

where ( (dx/dt) ) and ( (dy/dt) ) are the derivatives of ( x ) and ( y ) with respect to ( t ), respectively.

Given ( x = t - 3t^2 ) and ( y = t^2 - t ), we find:

[ \frac{dx}{dt} = 1 - 6t ] [ \frac{dy}{dt} = 2t - 1 ]

Now, substitute these derivatives into the arc length formula and integrate over the interval ([1,2]):

[ L = \int_{1}^{2} \sqrt{(1 - 6t)^2 + (2t - 1)^2} , dt ]

[ L = \int_{1}^{2} \sqrt{(1 - 6t)^2 + (2t - 1)^2} , dt ]

[ L = \int_{1}^{2} \sqrt{(1 - 6t)^2 + (2t - 1)^2} , dt ]

Now, you can integrate this expression to find the arc length ( L ).

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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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