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

What is the arclength of #(t-1,t/(t+5))# on #t in [-1,1]#?

Answer 1

Aurora Alvarado

The arc length is approximately #2.05# units.

The arc length of parametric functions is

#int_(-1)^1 sqrt((dy/(dt))^2 + ((dx)/(dt))^2) dt#

#int_(-1)^1 sqrt(1 + ((t + 5 - t)/(t + 5)^2)^2) dt#

#int_(-1)^1 sqrt(1 + 25/(t + 5)^4) dt#

Use a calculator to evaluate this tricky integral. Thus

#I = 2.05#

Hopefully this helps!

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Answer 2

Cameron Alexander

To find the arc length of the curve ( (t-1, \frac{t}{t+5}) ) on the interval ( [-1, 1] ), you would integrate the square root of the sum of the squares of the first derivative of each component function with respect to ( t ) over the interval.

The first derivative of ( t-1 ) is ( 1 ), and the first derivative of ( \frac{t}{t+5} ) is ( \frac{5}{(t+5)^2} ).

So, the arc length integral becomes:

[ \int_{-1}^{1} \sqrt{1 + \left(\frac{5}{(t+5)^2}\right)^2} , dt ]

Solving this integral will give you the arc length of the curve on the given interval.

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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.