If $\displaystyle F(x)= (3x²y, 2x-y)$, find

$\displaystyle

\int\limits_\gamma F(x)\,dx

$

where $\displaystyle \gamma$ is the directed path from (0,0) to (1,1) along the graph of the vector equation:

$\displaystyle x = (\sin t, 2t/\pi), (0 \leq t \leq \pi/2) $

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Here's what I did:

I recognized that the integral is not independent of path... So set up:

$\displaystyle \int M dx + N dy$

which gives me the sum of 4 integral terms... I evaluated these up to the following step so somebody could double check:

$\displaystyle 6/\pi [\pi/2 -1] - 6/\pi[\pi/3 - 7/9] + 4/\pi - 1/2$

Now does this answer seem correct to you?