Line Integral along a Parabola

[Char. Limit]

Homework Statement

a. Find a parametric equation to describe a parabola from the point (1,1) to the point (2,4).

b. Evaluate the line integral \int_C x ds along the parabolic segment in part a.

Homework Equations

\int_C x ds = \int_{t1}^{t2} x(t) |r'(t)| dt

The Attempt at a Solution

Well, for part a, the parametric equation r(t)=<t,t²>, 1≤t≤2, seemed to suffice. So I used that.

For part b, first I found r'(t) and got \sqrt{1+4 t^2}. Since x(t) = t, I then plugged this into my equation to get my new integral...

\int_1^2 t \sqrt{1+4 t^2} dt

Then I used the transform u=1+4t², du = 8t dt to transform my integral to...

\frac{1}{8} \int_{t=1}^{t=2} \sqrt{u} du = \frac{1}{8} \int_{u=5}^{u=17} \sqrt{u} du

Which I then evaluated to get...

\frac{1}{8} \frac{2}{3} \left[ u^{\frac{3}{2}} \right]_5^{17}

Which seems to equal...

\frac{17^{\frac{3}{2}} - 5^{\frac{3}{2}}}{12}

My question is... is this right?

 

[vela]

Looks fine to me.