Hi, I'm having trouble understanding how it is possible that one can integrate a parametric equation $\displaystyle y=g(t); x=h(t)$ when y is not a function of x.

for instance $\displaystyle x=2\cost-\cos2t, y=2sint-sin2t$

As I understand $\displaystyle \int_a^bf(x)dx=\int_{\alpha}^{\beta}g(t)h'(t)dt$ where both integrals measure the area under the curve.

But somehow, even if no f(x) exists, one can still use the parametric integral to measure the area within the shape it describes.

Can somebody explain to me why that is?