Surface Area with Parametric Equations — Question 7

Question 7

Problem

A cylinder is generated parametrically by rotating the horizontal segment $x=t,y=h$, $0\le t\le L$, about the $x$-axis, where $h,L>0$. Derive its lateral area.

Question 7 – Solution

Solution

1. Differentiate the horizontal segment: $$x'(t)=1, \qquad y'(t)=0.$$ Hence $$ds=\sqrt{1^2+0^2}\,dt=dt.$$

2. Every point of the segment is the constant distance $h$ from the $x$-axis, assuming $h\ge0$. Thus the radius of rotation is $h$.

3. Apply the surface-area formula: $$\begin{aligned} S&=2\pi\int_0^Lh\,dt\\ &=2\pi h[t]_0^L =\boxed{2\pi hL}. \end{aligned}$$

4. This is circumference times length: $(2\pi h)L$, which is precisely the lateral area of a cylinder of radius $h$ and length $L$.