# Thread: Compute the line integral

1. ## Compute the line integral

Calculate line integral $\int_c xydx + x^2y^3dy$, where c is triangle with vertices at points (0,0), (1,0) and (1,2).

How can I find r(t) ?

2. Originally Posted by Apprentice123
Calculate line integral $\int_c xydx + x^2y^3dy$, where c is triangle with vertices at points (0,0), (1,0) and (1,2).

How can I find r(t) ?
you have to state what direction you are going in. is it from (0,0) to (1,0) to (1,2)??

you will have to do three separate integrals, one for each side of the triangle. for each, your r(t) will be the vector function for the line.

Here is how to proceed:

Assuming we are going from (0,0) to (1,0) to (1,2), call the line connecting (0,0) to (1,0) $C_1$, the line connecting (1,0) and (1,2) $C_2$ and the line connecting (1,2) back to (0,0) $C_3$.

Then we have $\int_C F = \int_{C_1}F + \int_{C_2}F + \int_{C_3}F$.

lets concentrate on $C_1$.

for $C_1$, we have the line $y = 0$, with x ranging from 0 to 1. thus, we can parametrize the line by:

$x = t$, $y = 0$, for $0 \le t \le 1$

and so, $dx = 1$, $dy = 0$, and our integral for $C_1$ becomes:

$\int_0^1 t(0)(1) + t^2(0)^3(0)~dt = \int_0^1 0~dt = 0$

now do the same for $C_2$ and $C_3$.

(note, there is no need to introduce a $t$ here. we could have parametrized our line by $x = x$, $y = 0$ for $0 \le x \le 1$)

thank you