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 = ∫^b_a 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 = ∫^β_α 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 = ∫^β_α 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 = ∫_β^α g(t) f' (t) dt