Find arc length given #x=t\sint#, #y=t\cost# and #0\let\le1#?

I got up to #\int_0^1\sqrt(1+t^2)dt# using the parametric arc length formula #\color(indianred)(\int_a^b\sqrt((\dotx)^2+(\doty)^2)dt#, but a solution I found online changes the bounds from #\int_0^1# to #\color(palevioletred)(\int_0^(\tan^-1(1))#...

I assume the 0 is because #\tan^-1(0)=0#, but why arctangent? Is it because #t=\tan\theta# from trigonometric substitution?

Answer 1

#L=1/2(sqrt2+ln(1+sqrt2))# units.

#x=tsint# #x'=sint+tcost#
#y=tcost# #y'=cost-tsint#

Arc length is given by:

#L=int_0^1sqrt((sint+tcost)^2+(cost-tsint)^2)dt#

Expand and simplify:

#L=int_0^1sqrt(1+t^2)dt#
Apply the substitution #t=tantheta#:
#L=int_0^(tan^(-1)(1))sec^3thetad theta#

This is a known integral. If you do not have it memorized look it up in a table of integrals or apply integration by parts:

#L=1/2[secthetatantheta+ln|sectheta+tantheta|]_0^(tan^(-1)(1))#

Insert the limits of integration:

#L=1/2(sqrt2+ln(1+sqrt2))#
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Answer 2

To find the arc length of the curve defined by the parametric equations ( x = t \sin(t) ) and ( y = t \cos(t) ) for ( 0 \leq t \leq 1 ), you can use the formula for arc length for parametric equations:

[ s = \int_{a}^{b} \sqrt{\left(\frac{{dx}}{{dt}}\right)^2 + \left(\frac{{dy}}{{dt}}\right)^2} , dt ]

Substitute the given expressions for ( x ) and ( y ), and then differentiate them with respect to ( t ) to find ( \frac{{dx}}{{dt}} ) and ( \frac{{dy}}{{dt}} ).

[ \frac{{dx}}{{dt}} = \sin(t) + t\cos(t) ] [ \frac{{dy}}{{dt}} = \cos(t) - t\sin(t) ]

Then plug these into the formula:

[ s = \int_{0}^{1} \sqrt{(\sin(t) + t\cos(t))^2 + (\cos(t) - t\sin(t))^2} , dt ]

After simplifying the expression inside the square root and integrating over the interval ( 0 \leq t \leq 1 ), you will find the arc length of the curve.

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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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