Surface Area with Parametric Equations — Question 1

Question 1

Problem

Rotate the line segment $x=t$, $y=1-t$, $0\le t\le1$, about the $x$-axis. Find the surface area and identify the surface.

Question 1 – Solution

Solution

1. Differentiate the coordinates: $$x'(t)=1, \qquad y'(t)=-1.$$ Therefore, $$ds=\sqrt{1^2+(-1)^2}\,dt=\sqrt2\,dt.$$

2. The distance from $(x,y)$ to the $x$-axis is $y=1-t$, which is nonnegative on $0\le t\le1$.

3. Apply the surface-area formula: $$\begin{aligned} S&=2\pi\int_0^1 y\,ds\\ &=2\pi\sqrt2\int_0^1(1-t)\,dt\\ &=2\pi\sqrt2\left[t-\frac{t^2}{2}\right]_0^1 =\boxed{\pi\sqrt2}. \end{aligned}$$

4. The segment runs from $(0,1)$ to $(1,0)$. Rotating it about the $x$-axis produces a right circular cone with radius $1$, height $1$, and slant height $\sqrt2$. The result agrees with $\pi r\ell=\pi\sqrt2$.