Line integral across a vector field

[clandarkfire]

Homework Statement

$$\int_C \mathbf F\cdot d \mathbf r$$ where $$\mathbf F = x^2\vec{i}+e^{\sin^4{y}}\vec{j}$$
and C is the segment of y=x^2 from (-1,1) to (1,1).

Homework Equations

$$\int_C \mathbf F\cdot d \mathbf r=\int_a^b \mathbf F( \mathbf r(t))\cdot r'(t) dt=\int_C Pdx+Qdy$$ where $$\mathbf F = P\vec{i}+Q\vec{j}$$

The Attempt at a Solution

I parametrize C:
$$\mathbf r(t)=<t,t^2>|-1\leq t \leq 1$$
Well, I know how to apply the equations above as well as Green's theorem (I use this by letting C2 be a line from (1,1) to (-1,1) and integrating across the enclosed disk D), but no matter what I do, I find myself having to integrate some version of $$e^{\sin^4{y}}$$
There has to be some trick I'm missing. Could someone point me in the right direction?

 

[SammyS]

Homework Statement

$$\int_C \mathbf F\cdot d \mathbf r$$ where $$\mathbf F = x^2\vec{i}+e^{\sin^4{y}}\vec{j}$$
and C is the segment of y=x^2 from (-1,1) to (1,1).

Homework Equations

$$\int_C \mathbf F\cdot d \mathbf r=\int_a^b \mathbf F( \mathbf r(t))\cdot r'(t) dt=\int_C Pdx+Qdy$$ where $$\mathbf F = P\vec{i}+Q\vec{j}$$

The Attempt at a Solution

I parametrize C:
$$\mathbf r(t)=<t,t^2>|-1\leq t \leq 1$$
Well, I know how to apply the equations above as well as Green's theorem (I use this by letting C2 be a line from (1,1) to (-1,1) and integrating across the enclosed disk D), but no matter what I do, I find myself having to integrate some version of $$e^{\sin^4{y}}$$
There has to be some trick I'm missing. Could someone point me in the right direction?

If $\displaystyle \mathbf r(t)=<t,\,t^2>\,,\$ then $\displaystyle d\mathbf r(t)=<1,\,2t>dt\ .$

You will have an odd integrand involving the problematic $\displaystyle e^{\sin^4{y}}\ .$

 

[clandarkfire]

Does this get me anywhere? It seems to still leave me with something unsolveable.

 

[LCKurtz]

Check for independence of path and use the straight line from (-1,1) to (1,1).