Surface Area of Revolution
When a curve \(y=y(x)\) is revolved around an axis, it sweeps out a surface. The surface area is found by
summing the areas of many thin “bands” (like truncated cones) and taking a limit, which produces an integral.
1) Main formula (explicit function)
Around the x-axis (radius is the distance to the x-axis, \(r(x)=|y(x)|\)):
\[
S = 2\pi\int_a^b |y(x)|\sqrt{1+\left(\frac{dy}{dx}\right)^2}\,dx
\]
Around the y-axis (radius is the distance to the y-axis, \(r(x)=|x|\)):
\[
S = 2\pi\int_a^b |x|\sqrt{1+\left(\frac{dy}{dx}\right)^2}\,dx
\]
The factor \(\sqrt{1+(dy/dx)^2}\) is the “stretch” from converting a small horizontal change \(dx\) into the
arc-length element \(ds\), since \(ds=\sqrt{1+(dy/dx)^2}\,dx\).
2) Frustum approximation (piecewise cones)
If you split \([a,b]\) into \(N\) equal pieces of width \(\Delta x\), each small segment of the curve revolved
around the axis forms a truncated cone (a frustum). Its lateral area is:
\[
\Delta S_i \approx \pi\,(r_i+r_{i+1})\,s_i,
\qquad
s_i=\sqrt{(\Delta x)^2+(\Delta y)^2}.
\]
Summing over all segments gives the numeric approximation:
\[
S \approx \sum_{i=0}^{N-1}\pi\,(r_i+r_{i+1})\,s_i.
\]
This method is usually very robust even when \(\frac{dy}{dx}\) becomes very large near an endpoint,
because it works directly with \(\Delta x\) and \(\Delta y\).
3) Volume pair (sanity check)
Many problems naturally pair surface area with a volume of revolution.
This calculator can compute a matching volume for comparison:
Around the x-axis (disks):
\[
V = \pi\int_a^b \big(y(x)\big)^2\,dx
\]
Around the y-axis (shells):
\[
V = 2\pi\int_a^b |x|\,|y(x)|\,dx
\]
4) Worked example
Surface of \(y=\sqrt{x}\) from \(0\) to \(1\) around the x-axis.
Step 1: derivative.
\[
y=\sqrt{x}=x^{1/2}
\quad\Rightarrow\quad
\frac{dy}{dx}=\frac{1}{2\sqrt{x}}.
\]
Step 2: substitute into the surface area formula.
\[
\begin{aligned}
S
&=2\pi\int_0^1 \sqrt{x}\,\sqrt{1+\left(\frac{1}{2\sqrt{x}}\right)^2}\,dx \\
&=2\pi\int_0^1 \sqrt{x}\,\sqrt{1+\frac{1}{4x}}\,dx \\
&=2\pi\int_0^1 \sqrt{x+\frac14}\,dx.
\end{aligned}
\]
Step 3: evaluate (exact form shown; the calculator also gives a numeric result).
\[
\begin{aligned}
S
&=2\pi\left[\frac{2}{3}\left(x+\frac14\right)^{3/2}\right]_0^1 \\
&=\frac{4\pi}{3}\left(\left(\frac54\right)^{3/2}-\left(\frac14\right)^{3/2}\right)
\approx 5.3304.
\end{aligned}
\]
5) Advanced extension (parametric)
For parametric curves \(x=x(t)\), \(y=y(t)\), a common extension is
\(ds=\sqrt{(dx/dt)^2+(dy/dt)^2}\,dt\), and the radius is the distance to the chosen axis.
The same “band” idea applies, and the calculator’s frustum approach mirrors that geometry.
