The Central Idea
A surface of revolution is generated by rotating a curve around a fixed line called the axis of revolution.
Every point on the original curve moves in a circle around the axis.
The collection of all these circular paths forms a surface.
Symbolically:
\[ \boxed{\text{curve}+\text{rotation}=\text{surface of revolution}}. \]The Generating Curve
The original planar curve is often called the generating curve, profile curve, or meridian curve.
Its distance from the axis determines the radius of the circular path traced during rotation.
Its position along the axis determines where that circle lies in space.
The Axis of Revolution
The axis of revolution remains fixed while the generating curve rotates.
Common choices are:
- the \(x\)-axis;
- the \(y\)-axis;
- a vertical or horizontal translated line;
- another line in the plane.
The same generating curve can produce different surfaces when rotated around different axes.
Rotation Around the \(x\)-Axis
Suppose a curve is given by
\[ y=f(x), \]with
\[ f(x)\ge0. \]Rotating the point
\[ (x,f(x)) \]around the \(x\)-axis produces a circle of radius
\[ f(x). \]The resulting surface is organized by circular cross-sections perpendicular to the \(x\)-axis.
Implicit Equation for Revolution About the \(x\)-Axis
In the original plane, the vertical distance from the \(x\)-axis is
\[ y. \]After rotation into three dimensions, radial distance from the \(x\)-axis becomes
\[ \sqrt{y^2+z^2}. \]Therefore, if
\[ y=f(x), \]then a corresponding surface equation is
\[ \boxed{y^2+z^2=[f(x)]^2}. \]Example: A Cylinder from a Horizontal Line
Begin with the line
\[ y=R, \]where \(R>0\).
Rotate it around the \(x\)-axis.
Since every point remains distance \(R\) from the axis, the resulting surface is a circular cylinder:
\[ \boxed{y^2+z^2=R^2}. \]Example: A Cone from a Line Through the Origin
Consider
\[ y=mx, \]over a suitable interval.
Rotating around the \(x\)-axis gives
\[ y^2+z^2=m^2x^2. \]Thus the generated surface is a cone:
\[ \boxed{y^2+z^2=m^2x^2}. \]A Double Cone
If the generating line extends through both positive and negative \(x\)-values, the equation
\[ y^2+z^2=m^2x^2 \]describes two nappes meeting at the origin.
This is the same kind of double cone used in the geometric definition of conic sections.
Example: A Sphere from a Semicircle
Consider the upper semicircle
\[ y=\sqrt{R^2-x^2}, \qquad -R\le x\le R. \]Rotating it around the \(x\)-axis gives
\[ y^2+z^2=R^2-x^2. \]Rearranging:
\[ \boxed{x^2+y^2+z^2=R^2}. \]The surface is a sphere of radius \(R\).
Why a Semicircle Is Enough to Generate a Sphere
The upper semicircle already contains one point at each nonnegative distance from the axis required by the sphere.
Rotating that radius around the axis sweeps through the entire circular cross-section.
Rotating the full circle would cover the same sphere more than once.
Rotation Around the \(y\)-Axis
Suppose a curve is written
\[ x=g(y), \]with
\[ g(y)\ge0. \]Rotating the curve around the \(y\)-axis produces circular cross-sections whose radii are
\[ g(y). \]Implicit Equation for Revolution About the \(y\)-Axis
Radial distance from the \(y\)-axis is
\[ \sqrt{x^2+z^2}. \]Therefore, rotating
\[ x=g(y) \]around the \(y\)-axis gives
\[ \boxed{x^2+z^2=[g(y)]^2}. \]Example: Paraboloid of Revolution
Consider the parabola
\[ y=x^2. \]Rewrite it as
\[ x^2=y. \]Rotating around the \(y\)-axis replaces \(x^2\) by the squared radial distance
\[ x^2+z^2. \]The result is
\[ \boxed{x^2+z^2=y}. \]This is a paraboloid of revolution.
Profiles and Cross-Sections
A surface of revolution has two especially important families of cross-sections.
- Meridian sections contain the axis of revolution and reproduce the profile geometry.
- Parallel sections perpendicular to the axis are circles, points, or empty.
Meridians
A meridian is a curve obtained by intersecting a surface of revolution with a plane containing the axis.
The original generating curve is one such meridian.
Because the surface is rotationally symmetric, rotating that meridian around the axis generates all other meridian sections.
Parallels
A parallel is obtained by intersecting the surface with a plane perpendicular to the axis of revolution.
For an ordinary surface of revolution, a nondegenerate parallel is a circle.
Its radius is the distance from the corresponding profile point to the axis.
Rotational Symmetry
A surface of revolution is invariant under rotation about its axis.
If the surface is rotated through any angle
\[ \theta, \]around that axis, the point set remains unchanged.
This continuous rotational symmetry is one of the defining geometric features of the surface.
Parametric Representation About the \(x\)-Axis
Suppose
\[ y=f(u) \]and \(x=u\).
Rotation through angle \(v\) around the \(x\)-axis gives
\[ \boxed{ \mathbf r(u,v) = \langle u,\, f(u)\cos v,\, f(u)\sin v \rangle }. \]Typically,
\[ 0\le v<2\pi. \]Why a Surface Needs Two Parameters
A curve is one-dimensional and can generally be traced using one parameter.
A surface is two-dimensional and generally requires two independent parameters.
For a surface of revolution:
- one parameter moves along the generating curve;
- the other rotates around the axis.
Parametric Sphere
A sphere of radius \(R\) can be written
\[ \boxed{ \mathbf r(u,v) = \langle R\cos u,\, R\sin u\cos v,\, R\sin u\sin v \rangle }. \]One convenient domain is
\[ 0\le u\le\pi, \qquad 0\le v<2\pi. \]Here \(u\) moves along a semicircular profile and \(v\) performs the revolution.
Parametric Cylinder
A cylinder of radius \(R\) can be represented as
\[ \boxed{ \mathbf r(u,v) = \langle u,\, R\cos v,\, R\sin v \rangle }. \]The parameter \(u\) moves along the cylinder's axis, while \(v\) moves around each circular cross-section.
Parametric Cone
A cone can be represented by
\[ \boxed{ \mathbf r(u,v) = \langle u,\, mu\cos v,\, mu\sin v \rangle }. \]Here the circular radius grows linearly with the axial parameter \(u\).
Rotation About a Translated Horizontal Axis
Suppose the axis of revolution is
\[ y=c \]rather than the \(x\)-axis.
A point \((x,y)\) has distance
\[ |y-c| \]from that axis.
That distance becomes the radius of the generated circular orbit.
Example: Rotation About \(y=2\)
Consider the line
\[ y=5. \]Its distance from the axis
\[ y=2 \]is
\[ 3. \]Rotating the line around \(y=2\) produces a cylinder of radius
\[ \boxed{3}. \]The Torus
A torus can be generated by rotating a circle around a coplanar axis that does not intersect the circle.
Let the generating circle have radius
\[ r \]and let its center be distance
\[ R \]from the axis, with
\[ R>r>0. \]The resulting surface has a ring-like form.
Parametric Torus
A standard torus around the \(z\)-axis has parametrization
\[ \boxed{ x=(R+r\cos u)\cos v, } \] \[ \boxed{ y=(R+r\cos u)\sin v, } \] \[ \boxed{ z=r\sin u. } \]Both parameters may run through
\[ 0\le u,v<2\pi. \]Major and Minor Radii of a Torus
In the torus formulas:
- \(R\) is the major radius, measuring from the axis of revolution to the center of the generating circle;
- \(r\) is the minor radius, the radius of the generating circle itself.
These two scales control the overall ring size and tube thickness.
Different Torus-Like Cases
The relationship between \(R\) and \(r\) changes the geometry.
- \(R>r\): ring torus;
- \(R=r\): horn torus;
- \(0
The familiar doughnut-shaped torus is the ring-torus case.
Not Every Rotated Curve Produces an Embedded Surface
If the generating curve crosses the axis or if the rotational sweep causes different parameter values to reach the same spatial points, the resulting parametrization can self-overlap.
Therefore, a surface of revolution may require care when interpreting self-intersections or repeated coverage.
Implicit and Parametric Descriptions
A surface of revolution can often be described in several ways.
For a cylinder:
\[ y^2+z^2=R^2 \]is implicit, while
\[ \mathbf r(u,v) = \langle u,R\cos v,R\sin v\rangle \]is parametric.
Both describe the same surface but emphasize different structure.
Why Parametric Form Is Especially Useful
Parametric form makes the generating mechanism visible.
It naturally records:
- position along the profile;
- rotation angle;
- surface orientation;
- tangent directions;
- inputs for surface-area calculations.
Surface Tangent Directions
For a parametrized surface
\[ \mathbf r(u,v), \]changing \(u\) while holding \(v\) fixed gives one tangent direction:
\[ \mathbf r_u. \]Changing \(v\) while holding \(u\) fixed gives another:
\[ \mathbf r_v. \]These two tangent vectors locally span the tangent plane when they are linearly independent.
Normal Direction to a Parametric Surface
When
\[ \mathbf r_u \]and
\[ \mathbf r_v \]are linearly independent, their cross product
\[ \boxed{\mathbf r_u\times\mathbf r_v} \]is perpendicular to the surface.
This gives a normal direction and prepares the way for later surface geometry.
Surface Area from Parametrization
For a sufficiently regular parametrized surface
\[ \mathbf r(u,v), \]the area element is
\[ \boxed{ dS= \|\mathbf r_u\times\mathbf r_v\|\,du\,dv }. \]Therefore, over a parameter domain \(D\),
\[ \boxed{ S= \iint_D \|\mathbf r_u\times\mathbf r_v\| \,du\,dv }. \]This is the general surface-area framework.
Surface Area of Revolution About the \(x\)-Axis
For
\[ y=f(x)\ge0 \]rotated around the \(x\)-axis from \(x=a\) to \(x=b\), the surface area is
\[ \boxed{ S= 2\pi \int_a^b f(x) \sqrt{1+[f'(x)]^2} \,dx }. \]The formula combines circumference with differential arc length.
Why the Surface-Area Formula Has This Form
A tiny arc segment of length
\[ ds \]at distance
\[ y \]from the axis sweeps approximately a narrow band with area
\[ 2\pi y\,ds. \]Since
\[ ds= \sqrt{1+[f'(x)]^2}\,dx, \]integration gives
\[ S= 2\pi\int y\,ds. \]Surface Area About the \(y\)-Axis
If
\[ x=g(y)\ge0 \]is rotated around the \(y\)-axis, then
\[ \boxed{ S= 2\pi \int g(y) \sqrt{1+[g'(y)]^2} \,dy }. \]Again, the radial distance from the axis multiplies the arc-length element.
Example: Lateral Area of a Cylinder
Rotate
\[ y=R \]from
\[ x=0 \]to
\[ x=h \]around the \(x\)-axis.
Since
\[ f'(x)=0, \]the area is
\[ S= 2\pi\int_0^h R\,dx. \]Therefore,
\[ \boxed{S=2\pi Rh}. \]This is the cylinder's lateral surface area.
Example: Surface Area of a Sphere
Rotate the semicircle
\[ y=\sqrt{R^2-x^2} \]about the \(x\)-axis.
The resulting surface is a sphere, whose total surface area is
\[ \boxed{4\pi R^2}. \]This result can be derived from the surface-of-revolution formula.
Surface Area Is Different from Volume
Rotating a curve creates a surface.
Rotating a region can create a solid.
Therefore:
\[ \boxed{\text{curve}\rightarrow\text{surface}} \]while
\[ \boxed{\text{region}\rightarrow\text{solid}}. \]Surface area and volume are different measurements of these different objects.
Surface of Revolution Versus Solid of Revolution
Suppose
\[ y=f(x) \]is rotated around the \(x\)-axis.
Rotating only the graph produces the boundary surface.
Rotating the entire region between the graph and axis can produce a solid.
This distinction is essential in calculus and spatial geometry.
Cross-Sections Reveal the Surface
A three-dimensional surface can often be understood by slicing it with planes.
For a surface of revolution, cross-sections perpendicular to the axis are especially simple because they are circles.
Cross-sections through the axis reproduce or reflect the generating profile.
Example: Cross-Sections of a Sphere
Consider
\[ x^2+y^2+z^2=R^2. \]Fix
\[ x=c. \]Then
\[ y^2+z^2=R^2-c^2. \]This is a circle of radius
\[ \boxed{\sqrt{R^2-c^2}}. \]Example: Cross-Sections of a Paraboloid
Consider
\[ x^2+z^2=y. \]Fix
\[ y=c. \]Then
\[ x^2+z^2=c. \]For \(c>0\), the cross-section is a circle of radius
\[ \boxed{\sqrt c}. \]At \(c=0\), the circle collapses to the vertex.
Surface Profiles as Functions
A useful surface of revolution can be understood by tracking a radius as a function of axial position.
For revolution around the \(x\)-axis:
\[ \boxed{r(x)=|f(x)|}. \]This radius function determines every perpendicular circular cross-section.
Why the Absolute Value Can Matter
Distance from an axis is nonnegative.
If a generating graph has
\[ f(x)<0, \]its distance from the \(x\)-axis is
\[ |f(x)|. \]Rotating \(y=f(x)\) and \(y=-f(x)\) about the \(x\)-axis produces the same surface wherever the magnitudes agree.
Redundant Generating Curves
Because opposite points may sweep out the same circular orbit, a generating curve can sometimes cover the same surface more than once.
For example, rotating both the upper and lower semicircles around the \(x\)-axis generates the same sphere twice.
Efficient parametrizations avoid unnecessary duplicate coverage when possible.
Surfaces of Revolution and Transformations
Revolution is a continuous family of rotations.
In matrix language, rotating the point
\[ (x,y,0) \]around the \(x\)-axis by angle \(\theta\) gives
\[ \boxed{ \begin{pmatrix} x\\ y\cos\theta\\ y\sin\theta \end{pmatrix} }. \]This is the linear-algebraic foundation of the parametric formula for revolution.
Rotation Matrix About the \(x\)-Axis
The three-dimensional rotation matrix is
\[ \boxed{ R_x(\theta)= \begin{pmatrix} 1&0&0\\ 0&\cos\theta&-\sin\theta\\ 0&\sin\theta&\cos\theta \end{pmatrix} }. \]Applying it to
\[ \begin{pmatrix} x\\ y\\ 0 \end{pmatrix} \]gives
\[ \begin{pmatrix} x\\ y\cos\theta\\ y\sin\theta \end{pmatrix}. \]Surfaces of Revolution and Coordinate Systems
Cylindrical and spherical coordinates are naturally adapted to rotational symmetry.
For example, the cylinder
\[ x^2+y^2=R^2 \]becomes simply
\[ \boxed{r=R} \]in cylindrical coordinates.
Choosing coordinates that match symmetry can greatly simplify a surface.
Common Error: Rotating a Point Instead of the Entire Curve
One point traces one circle.
A surface of revolution requires every point of the generating curve to rotate.
The surface is the union of all those circular paths.
Common Error: Confusing the Axis with a Coordinate Axis
The axis of revolution does not have to be
\[ x=0 \]or
\[ y=0. \]A translated line such as
\[ y=3 \]can also serve as the axis.
The relevant radius is always the perpendicular distance to the chosen axis.
Common Error: Forgetting That Radius Must Be Nonnegative
In a surface of revolution, radius is a distance.
Therefore, a graph value such as
\[ y=-4 \]represents a rotational radius of
\[ 4 \]about the \(x\)-axis.
Common Error: Confusing Surface Area with Cross-Sectional Area
A circular cross-section of radius \(r\) has area
\[ \pi r^2. \]But a narrow surface band is related to circumference:
\[ 2\pi r. \]That is why surface-of-revolution formulas involve
\[ 2\pi r\,ds \]rather than
\[ \pi r^2. \]Common Error: Ignoring Duplicate Coverage
A parametrization may sweep through the same surface points more than once.
For example, letting the rotation parameter cover
\[ 0\le v\le4\pi \]traces two full revolutions.
The point set is unchanged, but the parametrization is redundant.
Common Error: Assuming Every Surface with Circular Sections Is a Surface of Revolution
Circular cross-sections alone do not automatically guarantee rotational symmetry about one fixed axis.
The circles must be arranged consistently around a common axis for the surface to qualify as a surface of revolution.
A Reliable Procedure for Building a Surface of Revolution
- Identify the generating curve.
- Identify the axis of revolution.
- Measure the perpendicular distance from a profile point to the axis.
- Interpret that distance as the radius of a circular orbit.
- Introduce a rotation angle parameter.
- Write a parametric or implicit surface equation.
- Choose parameter intervals that cover the intended surface.
- Check for duplicate coverage or self-intersection.
- Use cross-sections to verify the geometry.
- Apply a surface-area formula if measurement is required.
Example: Full Surface Construction
Rotate
\[ y=x^2, \qquad 0\le x\le2, \]around the \(x\)-axis.
The radial distance from the axis is
\[ x^2. \]Introduce a rotation angle \(v\):
\[ \boxed{ \mathbf r(u,v) = \langle u,\, u^2\cos v,\, u^2\sin v \rangle }, \]with
\[ 0\le u\le2, \qquad 0\le v<2\pi. \]The corresponding implicit relation is
\[ \boxed{y^2+z^2=x^4}, \]restricted to
\[ 0\le x\le2. \]Practice 1: Cylinder
Rotate
\[ y=4 \]around the \(x\)-axis.
Write the resulting implicit surface equation.
Practice 2: Cone
Rotate
\[ y=2x \]around the \(x\)-axis.
Write the resulting surface equation.
Practice 3: Sphere
Rotate
\[ y=\sqrt{25-x^2}, \qquad -5\le x\le5, \]around the \(x\)-axis.
Identify the surface.
Practice 4: Paraboloid
Rotate
\[ x=\sqrt y, \qquad y\ge0, \]around the \(y\)-axis.
Write the implicit surface equation.
Practice 5: Parametric Cylinder
Write a parametrization of a cylinder of radius \(3\) whose axis is the \(x\)-axis.
Practice 6: Cross-Section
For the sphere
\[ x^2+y^2+z^2=36, \]find the radius of the circular cross-section at
\[ x=2. \]Practice 7: Lateral Cylinder Area
A cylinder has radius
\[ R=3 \]and height
\[ h=10. \]Find its lateral surface area.
Practice 8: Torus Radii
In
\[ x=(7+2\cos u)\cos v, \] \[ y=(7+2\cos u)\sin v, \] \[ z=2\sin u, \]identify the major radius and minor radius.
Practice 9: Surface or Solid?
A graph curve is rotated around an axis.
Does this operation by itself generate a surface or a solid?
Practice 10: Radius from a Translated Axis
The line
\[ y=9 \]is rotated around
\[ y=4. \]What is the radius of the resulting cylinder?
Practice Answers
Practice 1
The radius is \(4\), so
\[ \boxed{y^2+z^2=16}. \]Practice 2
Replace the squared distance from the \(x\)-axis by
\[ y^2+z^2. \]Since the generating line gives radius \(2x\),
\[ \boxed{y^2+z^2=4x^2}. \]Practice 3
Rotation gives
\[ y^2+z^2=25-x^2. \]Therefore,
\[ \boxed{x^2+y^2+z^2=25}. \]The surface is a sphere of radius \(5\).
Practice 4
Since
\[ x^2=y, \]rotation around the \(y\)-axis gives
\[ \boxed{x^2+z^2=y}. \]Practice 5
One parametrization is
\[ \boxed{ \mathbf r(u,v) = \langle u,\, 3\cos v,\, 3\sin v \rangle }. \]Practice 6
Set
\[ x=2. \]Then
\[ y^2+z^2=36-4=32. \]The radius is
\[ \boxed{\sqrt{32}=4\sqrt2}. \]Practice 7
Use
\[ S=2\pi Rh. \]Therefore,
\[ S=2\pi(3)(10)=\boxed{60\pi}. \]Practice 8
The major radius is
\[ \boxed{R=7}, \]and the minor radius is
\[ \boxed{r=2}. \]Practice 9
Rotating a curve generates a
\[ \boxed{\text{surface}}. \]Rotating a region may generate a solid.
Practice 10
The distance between the two horizontal lines is
\[ |9-4|=5. \]Therefore, the cylinder radius is
\[ \boxed{5}. \]What to Remember
A surface of revolution is produced by rotating a planar profile around a fixed axis.
- Rotation of \(y=f(x)\) about the \(x\)-axis gives \[ y^2+z^2=[f(x)]^2. \]
- Rotation of \(x=g(y)\) about the \(y\)-axis gives \[ x^2+z^2=[g(y)]^2. \]
- A convenient parametrization about the \(x\)-axis is \[ \mathbf r(u,v) = \langle u,\, f(u)\cos v,\, f(u)\sin v \rangle. \]
- One parameter moves along the profile; the other rotates around the axis.
- Perpendicular cross-sections are circular.
- Surface area about the \(x\)-axis is \[ S= 2\pi \int f(x)\sqrt{1+[f'(x)]^2}\,dx. \]
The central geometric habit is: track the distance from the generating curve to the axis, because that distance becomes the radius of rotation.
Where We Go Next
Surfaces of revolution extend planar curves into three dimensions through symmetry. The next lesson broadens this idea to a major family of surfaces defined directly by second-degree equations.
In Quadratic Surfaces, we will study ellipsoids, paraboloids, hyperboloids, cones, cylinders, and the way their equations and cross-sections determine spatial form.