Find the area of a parametric curve using Greens Theorem

Question

Given the curve $\vec\alpha(t)=(\cos(t),\sin(t)\cos(t))$ for $\frac {-\pi }2\le t \le \frac {\pi }2$

I am asked to find the volume so I need

$$\frac {\delta Q}{\delta x}-\frac {\delta P}{\delta y}=1$$ wich can be done with $\vec F=(y,2x)$ to get $\iint_C dA$

Now I am not sure on how to put the integral in terms of t, could someone explain?

Answer 1

Let's use $P=0$ and $Q = x$. Then, if we let $C = \partial \Omega$, then $$ \begin{split} \iint_\Omega dA &= \iint_\Omega \left(\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}\right)dA \\ &= \int_{\partial \Omega} P(x,y) dx + Q(x,y)dy \\ &= \int_C xdy \\ &= \int_{t=-\pi/2}^{t=\pi/2} x(t) dy(t) \\ &= \int_{t=-\pi/2}^{t=\pi/2} x(t) y'(t) dt \end{split} $$ Can you finish this?