 #1
zenterix
 66
 20
 Homework Statement:

Calculate the line integral with respect to arc length.
##\int_C (x+y)ds##, where ##C## is the triangle with vertices ##(0,0), (1,0)## and ##(0,1)##, traversed in a counterclockwise direction.
 Relevant Equations:

There are three separate paths on the triangle with parametric equations
$$\vec{r_1}(t)=\langle t,0 \rangle$$
$$\vec{r_2}(t)=\langle 1t,t \rangle$$
$$\vec{r_3}(t)=\langle 0,1t \rangle$$
The corresponding velocity vectors are
$$\vec{v_1}(t)=\langle 1,0 \rangle$$
$$\vec{v_2}(t)=\langle 1,1 \rangle$$
$$\vec{v_3}(t)=\langle 0,1 \rangle$$
and speeds
$$v_1(t)=1$$
$$v_2(t)=\sqrt{2}$$
$$v_3(t)=1$$
The line integral then becomes
$$\int_C (x+y)ds = \int_0^1 t dt+\int_0^1 \sqrt{2} dt+\int_0^1 (1t)dt$$
$$=1/2+\sqrt{2}+11/2$$
$$=\sqrt{2}+1$$
The answer at the end of the book says ##\sqrt{2}##.
Is this correct or is my solution correct?
Here is a depiction of the path where we are integrating
Is this correct or is my solution correct?
Here is a depiction of the path where we are integrating