# A vector equation thingy

#### Dr Zoidburg

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

$$\displaystyle x=t \Rightarrow \tfrac{\delta x}{\delta t} = 1$$
$$\displaystyle y= \tfrac{1}{t} \Rightarrow \tfrac{\delta y}{\delta t} = -\tfrac{1}{t^2}$$

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

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
Believe it or not, that is exactly right!(Clapping)

#### DrDank

Also try
$$\displaystyle \int_{C}f(X(t))\cdot X'(t) dt$$