# Cal III S reparametrization problem

crims0ned
Question:
reparametrize the curve r(t)=<cos(t)-tcos(t), sin(t)-tsin(t)> : t=0

I know I need to integrate the mag from zero to t but it's not working.

got down to the integral of sqrt((t^2)-2t+1) and after a u-sub and a theta sub I got the integral of sec(theta). Then got natural log(sec(theta)+tan(theta)) but if i back sub in I get ln(0) first of all and then a bunch of other nasty problems just i try to solve for t in terms of s.

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I actually got sqrt((t^2)-2*t+2)

the equation for the reparametrization should be s=diffident integral( sqrt((dx/dt)^2)+(dy/dt)^2).

I first get sqrt((sin(t)+(t-1)cos(t))^2 + (-sin(t)(1-t)-cos(t))^2)

= sqrt(((t^2)-2t+2)(sin(t)^2+cos(t)^2))........ sin(t)^2+cos(t)^2 becomes 1

now i'm left with strictly the integral of sqrt((t^2)-2t+2) evaluated from zero to t

I completed the square and got sqrt((t-1)^2 + 1) then u-subbed u=t-1

so integral sqrt(u^2 + 1) i then i used a trig sub to get to just the integral of sec^3(theta)

so now i have the integral of sec^3(theta) which equals (1/2)sqrt((t-1)^2 + 1)*(t-1) + (1/2)ln(t + sqrt((t-1)^2 + 1) - 1)

and I have to evaluate this from zero to t

...okay and here's my real problem. This question began and a reparametrization so I have to solve for t in terms of s. Other then this being some algebra I haven't worked in a while, I think I can solve it but is there a trig i.d. i missed in the beginning or something? because I don't think a s-parametrization should be this complicated, but maybe i'm wrong.

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The integral of sqrt(t^2-2*t+1) shouldn't be that hard if you notice t^2-2*t+1=(t-1)^2.

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Gold Member
Question:
reparametrize the curve r(t)=<cos(t)-tcos(t), sin(t)-tsin(t)> : t=0

I know I need to integrate the mag from zero to t but it's not working.

got down to the integral of sqrt((t^2)-2t+1) and after a u-sub and a theta sub I got the integral of sec(theta). Then got natural log(sec(theta)+tan(theta)) but if i back sub in I get ln(0) first of all and then a bunch of other nasty problems just i try to solve for t in terms of s.

$$\sqrt{t^2-2t+1}=\sqrt{(t-1)^2}=|t-1|=\left\{\begin{array}{lr}t-1, & t\geq1 \\ 1-t, & t \leq1\end{array}\right.$$

You shouldn't have much trouble integrating that. Edit Dick beat me to it Last edited:
Mentor
$$\sqrt{t^2-t+1}=\sqrt{(t-1)^2}=|t-1|=\left\{\begin{array}{lr}t-1, & t\geq1 \\ 1-t, & t \leq1\end{array}\right.$$

You shouldn't have much trouble integrating that. Edit Dick beat me to it Gabba2hey, you omitted a 2 in the first radical...
$$\sqrt{t^2-\bold{2}t+1}=\sqrt{(t-1)^2}=|t-1|=\left\{\begin{array}{lr}t-1, & t\geq1 \\ 1-t, & t \leq1\end{array}\right.$$

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Gabba2hey, you omitted a 2 in the first radical...
$$\sqrt{t^2-\bold{2}t+1}=\sqrt{(t-1)^2}=|t-1|=\left\{\begin{array}{lr}t-1, & t\geq1 \\ 1-t, & t \leq1\end{array}\right.$$

Good eye, just a typo though.

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I actually got sqrt((t^2)-2*t+2)

the equation for the reparametrization should be s=diffident integral( sqrt((dx/dt)^2)+(dy/dt)^2).

I first get sqrt((sin(t)+(t-1)cos(t))^2 + (-sin(t)(1-t)-cos(t))^2)

= sqrt(((t^2)-2t+2)(sin(t)^2+cos(t)^2))........ sin(t)^2+cos(t)^2 becomes 1

now i'm left with strictly the integral of sqrt((t^2)-2t+2) evaluated from zero to t

I completed the square and got sqrt((t-1)^2 + 1) then u-subbed u=t-1

so integral sqrt(u^2 + 1) i then i used a trig sub to get to just the integral of sec^3(theta)

so now i have the integral of sec^3(theta) which equals (1/2)sqrt((t-1)^2 + 1)*(t-1) + (1/2)ln(t + sqrt((t-1)^2 + 1) - 1)

and I have to evaluate this from zero to t

...okay and here's my real problem. This question began and a reparametrization so I have to solve for t in terms of s. Other then this being some algebra I haven't worked in a while, I think I can solve it but is there a trig i.d. i missed in the beginning or something? because I don't think a s-parametrization should be this complicated, but maybe i'm wrong.

It's better to simply add a new reply rather than edit your original response. If Mark hadn't replied, I never would have noticed your edited post (and I'm not sure Dick would have either), and you may never have gotten further replies.

I think its best if you post the entire original problem, word for word. (In a new reply!) So that we can see exactly what you are being asked to do.

I can't speak for Dick or Mark, but I'm not sure exactly what you mean be s-reparameterization. Is there some specific property that you would like your new paramater $s$ to satisfy, or is this just some random reparameterization?

crims0ned
we've been using s reparametrizations just as kind of random parametrization i guess. pretty much it seems to be finding the arc-length and evaluating it from some number (t sub not) to an indefinite t. And what ever we come up with we then solve for t is terms of s and substitute it back into the original equation.

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we've been using s reparametrizations just as kind of random parametrization i guess. pretty much it seems to be finding the arc-length and evaluating it from some number (t sub not) to an indefinite t. And what ever we come up with we then solve for t is terms of s and substitute it back into the original equation.

Do you mean that your new parameter $$s(t)[/itex] is supposed to represent the arclength from $\textbf{r}(0)$ to $\textbf{r}(t)$? So, [tex]s=\int_0^t ||\textbf{r}'(\overline{t})||d\overline{t}$$

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crims0ned
the equation is this $$s= \int \sqrt{(dx/dt)^2+(dy/dt)^2+(dz/dt)^2} dt$$ evaluated from some starting t to t

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Okay, and your original curve $\textbf{r}(t)$ is ____ ?

crims0ned
Yes, that's it, just the magnitude of dr/dt. Are you familiar with these types of problems? Because all of the ones I've worked pier to this and everything in my calculus book always out really clean.

crims0ned
the original problem is $$r(t)=<cos(t)-tcos(t), sin(t)-tsin(t)> ; t=o$$

crims0ned
there seems to be only an $$i$$ and $$j$$ component

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Okay, so you seem to have correctly gotten to the point where you are trying to evaluate

$$\int_{-1}^{t-1} \sqrt{u^2+1}du$$

Right?

From here, try using integration by parts once...what do you get?

crims0ned
once i got to $$\int_{-1}^{t-1} \sqrt{u^2+1}du$$ i trig subbed and got $$\int sec^3(\theta) d\theta$$ but can i change the limits along with the variable and still solve for $$s$$?

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once i got to $$\int_{-1}^{t-1} \sqrt{u^2+1}du$$ i trig subbed and got $$\int sec^3(\theta) d\theta$$ but can i change the limits along with the variable and still solve for $$s$$?

Well, if $u=\tan\theta$, then your limits on $\theta$ will be $\tan^{-1}(-1)$ and $\tan^{-1}(t-1)$.

Personally, I think that using integration by parts is easier than trying to integrate $\int\sec^3\theta d\theta$, but it's up to you.