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.