Home > Calculus > Determining the Length of a Parametric Curve (Parametric Form)

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

Answer 1

William Abdalla

Performed by WolframAlpha
#int_1^2 sqrt(4/t^2+(t+2 t log(t))^2) dt = 3.19515#

Let L be the parametric arclength, then the formula is:

#L = int_a^b sqrt((dx/dt)^2 + (dy/dt)^2)dt#

#x = t^2ln(t)# #dx/dt = 2tln(t) + t# #y = ln(t^2)# #dy/dt = 2/t# #a = 1# #b = 2#

Substitute the above into the formula:

#L = int_1^2 sqrt((2tln(t) + t)^2 + (2/t)^2)dt#

Integration performed by WolframAlpha

#int_1^2 sqrt(4/t^2+(t+2 t log(t))^2) dt = 3.19515#

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Answer 2

Charles Anctil

To find the arc length of the curve defined by the parametric equations (t^2ln(t), ln(t^2)) for t in the interval [1, 2], we use the formula for arc length:

L = ∫[a,b] √[ (dx/dt)^2 + (dy/dt)^2 ] dt

Substituting the parametric equations into the formula, we have:

L = ∫[1,2] √[ (d/dt(t^2ln(t)))^2 + (d/dt(ln(t^2)))^2 ] dt

After taking the derivatives and simplifying, we get:

L = ∫[1,2] √[ (2tln(t) + t)^2 + (2/t)^2 ] dt

Now, we integrate this expression over the interval [1,2] to find the arc length. This integral might not have a closed-form solution, so numerical methods or software may be used to approximate the value.

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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.