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

Answer 1
#f(t)=(t^2lnt,t-lnt)#
#f'(t)=(2tlnt+t,t-1/t)#
For a parametric function #f(t)=(x,y)#, the arc length for #tin[a,b]# is given by:
#s=int_a^bsqrt((dx/dt)^2+(dy/dt)^2)dt#

Thus, we have an arc length of:

#color(blue)(s=int_1^2sqrt((2tlnt+t)^2+(t-1/t)^2)dtapprox2.8914#

Find the approximation using a calculator or a website like WolframAlpha.

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Answer 2

To find the arc length of the curve ( (t^2 \ln t, t - \ln t) ) on the interval ([1, 2]), you can use the formula for arc length:

[ L = \int_{a}^{b} \sqrt{[f'(t)]^2 + [g'(t)]^2} , dt ]

where ( f(t) ) and ( g(t) ) are the parametric equations defining the curve. Calculate the derivatives of ( f(t) ) and ( g(t) ), then plug them into the formula and evaluate the integral over the interval ([1, 2]).

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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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