Two multivariable integral questions.

Question

1. Let $\mathbf{F}(x, y) = (-y^2, xy)$ and $C = \Bigl\{\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 : y \ge 0 \Bigr\}$. Determine $\displaystyle \int_C \mathbf{F} \cdot d\mathbf{x}$ if $C$ is oriented counter clockwise when viewed from $(0, 0, 1)$.

2. Determine $\displaystyle \int_C \mathbf{F} \cdot dx$ where $\mathbf{F}(x, y) = (x^2, -y)$ and $C$ is the graph of $ y = e^x$ from $(2, e^2)$ to $(0, 1)$.

For the first problem, I used the parametrization $g(t) = (acos(t), bsin(t))$, $0 \le t \le 2\pi$ and used the equation $\displaystyle \int_C \mathbf{F} \cdot d\mathbf{x} = \displaystyle \int_a^b \mathbf{F}(\mathbf{g}(t))\cdot \mathbf{g}'(t)dt$, but I got $\mathbf{F}(\mathbf{g}(t))\cdot \mathbf{g}'(t) = 2ab^2sin(t)cos^2(t)$ and now I'm stuck.

As for the second problem, I have no clue how to solve it. Don't know how to parametrize it and don't see how Green's theorem will work.

Answer 1

For the first one use a $\textbf{u}$-substitution with $\textbf{u} = \cos(t)$. For the second use the parametrization $\gamma(t) = (t,e^t)$ for the curve $C$ and solve like the way you did on problem $1$. Also for problem $1$, the limits should be from $0$ to $\pi$ since you want to use the parametrization of the ellipse;

$$\alpha(t) = \left(a \cos t, b \sin t\right)$$

Answer 2

For the second problem, a very useful parametrization is that of the curve $y=f(x),\ x\in[a,b]$ given by $\vec{r}(x) = x\vec{i}+f(x)\vec{j},\ a\leq x\leq b.$ Note that $x$ is the parameter here.