# Thread: line integral

1. ## line integral

vector field F= (yzcosx, zsinx, ysinx)
compute the integral over C where C is parameterized by p(t)= (t^t, t^2) for t [1,2]

we just learned the integral theorem dealing with the fundamental theorem of calculus. i'm not sure as to how to do this problem..i got something really complicated but the theorem is supposed to make it really simple. any suggestions? thanks so much

2. There are many integral theorems dealing with the fundamental theorem of calculus.

The line integral in a vector field is usually an integral of the dot product $\vec F\cdot d\vec r$. The version of the fundamental theorem of calculus which applies to line integrals in a vector field states that if $\vec F(p)$ is the gradient of some scalar field $G(p)$ then $\int_{p_1}^{p_2}\vec F\cdot d\vec r = G(p_2)-G(p_1)$. Try to set $\vec F$ equal to the gradient of some unknown scalar field $G$ and see if you can find $G$ and then apply the theorem. If you cannot find $G$ then you will need to evaluate the integral directly by using the given parametrization, and using that

$\int_{p_1}^{p_2}\vec F(\vec r)\cdot d\vec r = \int_{t_1}^{t_2}\vec F(\vec r(t))\cdot \vec r\ '(t) dt$

where $\vec r (t) \ \ (t_1 \leq t \leq t_2)$ is your parametrization.