Consider the non-self-intersecting plane curve defined by the parametric equations \(x=x(t)\) and \(y=y(t)\) for \(a \le t \le b\), and assume \(x(t)\) is differentiable.
Suppose we want to find the area under this curve between \(t=a\) and \(t=b\). Let \(x(a) = \alpha\) and \(x(b) = \beta\). Finding the area under the curve between \(t=a\) and \(t=b\) is the same as finding the area under curve between \(x = \alpha\) and \(x = \beta\):
\[A= \int_\alpha^\beta y \ dx \]
Since \(y=g(t)\) and \(\frac{dx}{dt} = x'(t)\):
\[A= \int_a^b y(t) x'(t)\ dt \]