I understand that the area under a parametric curve defined as $(x(t),y(t))$ from $a$ to $b$ can be found with $\int_a^b y(t) \space x'(t) \space dx$. What I don't understand is what happens to the area when the curve loops back around itself.
For example, I have a curve defined by $x(t) = \frac{\pi}{2} + t + cos(t)\cdot1.8$ and $y(t) = sin(t)+1$, and I want the area under this curve on $[-\frac{\pi}{2}, \frac{3\pi}{2}]$
When I try to find the area with this method, I get $0.2\pi$ If I tried to shade in the $0.2\pi$ on a drawing of this graph, where would I shade? What is the "area" that I am getting?
Also, if I replace the $1.8$ with a $2$, the area becomes $0$. Why is this?
The area that you compute is a closed self-intersecting shape (it is closed by a segment of the $x$ axis).
One of the parts is traversed clockwise and the other anticlockwise, so that the areas substract from one another, and the total can even vanish.
If you want the absolute area, you need to find the intersection point and compute the two areas separately.