Integrate r = $\sqrt{x^2+y^2}$ from (0,0) to (1,1) along the path (0,0) => (1,0) => (1,1)
My professor tells me to let $dr = dxi +dyj$ where $i $ and $j$ are the standard unit vectors. I don't really see how this is possible with a scalar function. I have been parametrizing:
$\int_Cr(x,y)dr = \int_{t_0}^tr(x_{(t)},y_{(t)})\sqrt{(\frac{dx}{dt})^2 + (\frac{dy}{dt})^2}dt$
and i get $C_1$ (0,0) => (1,0)
$x = t;dx/dt =1;y =0$
But my main confusion is when I try to parameterize C2: (1,0)=>(1,1) I get an integral which I cannot solve by any conventional analytical methods: x = 1, y = t, dy/dt = 1 $\int\sqrt{1 + t^2}dt$
So I have surely done something wrong, any help would be greatly appreciated.