A vector equation thingy

May 2008
77
3
Calculate \(\displaystyle \int_{C}f(X).dX\) where \(\displaystyle f(x,y)=(x^2+xy,y-x^2y)\) and C is parametrized by \(\displaystyle x=t, y= \frac{1}{t} \; (1\leq t \leq 3)\)

I'm not really sure how to go about this. This is what I've attempted:
\(\displaystyle x=t \Rightarrow \tfrac{\delta x}{\delta t} = 1\)
\(\displaystyle y= \tfrac{1}{t} \Rightarrow \tfrac{\delta y}{\delta t} = -\tfrac{1}{t^2}\)

\(\displaystyle \int_{C}f(X).dX = \int_{C} (x^2+xy).\partial x + (y-x^2y).\partial y = \int_{1}^{3} (t^2+1-\frac{1}{t^3}+\frac{1}{t}).\partial t \)

integrating that makes for a unpleasant mess: \(\displaystyle \frac{t^3}{3}+t+\frac{1}{2t^2}+ln(t)\) which is why I'm wondering if I'm doing this right.
 

HallsofIvy

MHF Helper
Apr 2005
20,249
7,909
Believe it or not, that is exactly right!(Clapping)
 
May 2010
20
8
Also try
\(\displaystyle \int_{C}f(X(t))\cdot X'(t) dt\)