Area with Parametric Equations
In this section we will find out a formula for ascertaining the area under a parametric curve specified by the parametric equations,
x = f (t)
y = g (t)
We will as well need to further add in the assumption that the curve is traced out precisely once as t increases from α to β.
We will do this in much similar way that we found the first derivative in the preceding section.
We will first remind how to find out the area under y = F(x) on a < x < b.
A = ∫ba F (x) dx
We will here think of the parametric equation x = f (t) as a substitution in the integral. We will as well assume that a = f(α) and b=f (β)) for the purposes of this formula. There is in fact no reason to assume that this will always be the case and so we will provide a corresponding formula later if it's the opposed case (b = f (α) and a = f (β)).
Thus, if this is going to be a substitution we'll require,
dx = f' (t) dt
Plugging this into the area formula on top of and making sure to change the limits to their corresponding t values provides us,
A = ∫βα F (f (t)) f' (t) dt
As we don't know what F(x) is we'll use the fact that
y = F (x)
= F (f (t)) = g (t)
and we reach at the formula that we want.
Area under Parametric Curve, Formula I
A = ∫βα g(t) f' (t) dt
Now, if we should happen to have b = f (α) and a = f (β) then the formula would be,
Area Under Parametric Curve, Formula II
A = ∫βα g(t) f' (t) dt