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

Arclength is given by:

Expand the squares and combine terms:

Complete the square in the square root:

This is a known integral:

Reverse the substitution:

Hence

Simplify:

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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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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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