Parametrization : $x = acos^3(t), y = asin^3(t)$ $a>0$
If you can solve it for me that would be awesome.:D If not, can you give me some hints? Tell me how to set it up and stuff. It's solveable by green's theorem allegedly. But I'm utterly lost. I still have a hard time understanding green's theorem and how the problems are set up.I mean I understand that you take the parital of the first function minus the second function partial, but usually I don't understand where they pull the vector fields from to apply to a problem like this that doesn't have a vector field
Plugging the vector field $\vec{F}(x,y)=y\vec{i}+-x\vec{j}$ into Green's Theorem, we achieve $\int_\vec{X}(y\vec{i}+-x\vec{j})d\vec{s}=\int\int_D 2 dx dy$. Therefore, the area of the region is $\frac{1}{2}\int_\vec{X}(y\vec{i}+-x\vec{j})d\vec{s}$. The reason that one chooses the vector field such is that this particular vector field makes the line integral become an area. Most vector fields do not have this property. And choosing a vector field out of thin air is one of those things, like treating derivatives as fractions, that make mathematicians uncomfortable but work anyways.