Line integral: cartesian to polar coordinates

Question

I am trying to calculate the line integral $\int_C (x^2+y^2) \,ds$ on $r=e^\theta$ over the interval $\theta=[0,\pi]$.

$\int_C (x^2+y^2) \,ds = \int_C r^2 \,ds$ leads me to $\int_0^\pi e^{3t} \,dt = \frac{1}{3} e^{3t} \Big|_0^\pi = \frac{1}{3}\big( e^{3\pi}-e^0 \big)\ldots$

The correct answer is $\dfrac{\sqrt{2} \cdot e^{3t}}{3}$. What am I doing wrong?

Answer 1

\begin{align} (x,y) & = (e^\theta \cos \theta, e^\theta \sin \theta)\\[8pt] (x',y') & = e^\theta (\cos \theta - \sin \theta, \cos\theta + \sin\theta) \ d\theta\\[8pt] ds & = \|(x',y')\| = e^\theta \sqrt 2 \ d\theta \end{align}

Or, staying in polar

arc length in polar coordinates

$$ds = \sqrt {r^2 + r'^2} \, d\theta$$