(923, #3) Evaluate the line integral by two methods: a) directly

(923, #25) Use exercise 24 to find the centroid of the triangle with vertices ( 0, 0 ), ( 1, 0 ), and ( 0, 1 ).
Solution: From exercise 24, we know

We are integrating on the following triangle:

On , the parametric equations are

So
and

On , the parametric equations are

So
and
On , the parametric equations are

So
and
Now we sum to get the solution over the path of

by C.A.H.