Surface Area with Parametric Equations
In this final section of looking at calculus applications with parametric equations we will take a look at determining the surface area of a region obtained by rotating a parametric curve about the x or y-axis.
We will rotate the parametric curve given by,
x = f (t)
y = g (t)
α ≤ t ≤ β
about the x or y-axis. We are going to suppose that the curve is traced out exactly one time as t increases from α to β. In fact at this point there isn't all that much to do. We know earlier that the surface area can be found by utilizing one of the following two formulas depending upon the axis of rotation.
S = ∫ 2Πy ds rotation about x- axis
S =∫ 2Πx ds rotation about y-axis
All that we required is a formula for ds to use and from the preceding section we have,
ds = √ [(dx/dt)2 + (dy/dt)2] dt
if x = f (t),
y = g(t),
α ≤ t ≤ β
which is exactly what we need.
We will require to be careful with the x or y that is in the original surface area formula. Back while we first looked at surface area we saw that occasionally we had to substitute for the variable in the integral and at another times we didn't. This was dependent on the ds which we used. However in this case, we will all time have to substitute for the variable. The ds that we use for parametric equations bring in a dt into the integral and meaning of this is that everything needs to be in terms of t. Hence, we will require to substitute the appropriate parametric equation for x or y depending upon the axis of rotation.