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Axis of Rotation

Axis of Rotation

A line about which a plane figure is rotated in three dimensional space to create a solid or surface.

 

 

See also

Solid of revolution, surface of revolution, torus, disk method, washer method, cylindrical shell method

Worked Example

Problem: A right triangle with legs of length 3 (along the x-axis from x = 0 to x = 3) and length 4 (along the y-axis from y = 0 to y = 4) is rotated about the x-axis. Describe the solid that is formed and identify the axis of rotation.
Step 1: Identify the axis of rotation. The triangle is rotated about the x-axis, so the axis of rotation is the line y = 0 (the x-axis itself).
Axis of rotation: y=0\text{Axis of rotation: } y = 0
Step 2: Determine what each point on the triangle does as it rotates. Every point at distance r from the x-axis traces a circle of radius r. The hypotenuse of the triangle connects (3, 0) to (0, 4), so its equation is:
y=−43x+4y = -\tfrac{4}{3}x + 4
Step 3: At each value of x from 0 to 3, the cross-section perpendicular to the x-axis is a disk whose radius equals the y-value on the hypotenuse. These stacked disks form a cone.
Radius at position x:  r(x)=−43x+4\text{Radius at position } x: \; r(x) = -\tfrac{4}{3}x + 4
Step 4: Compute the volume of the resulting cone using the disk method to confirm the shape. The cone has base radius 4 (at x = 0) and height 3 (along the x-axis).
V=π∫03(−43x+4)2dx=13π(4)2(3)=16πV = \pi \int_0^3 \left(-\tfrac{4}{3}x + 4\right)^2 dx = \tfrac{1}{3}\pi (4)^2(3) = 16\pi
Answer: The axis of rotation is the x-axis. Rotating the right triangle about this axis produces a cone with base radius 4, height 3, and volume 16π cubic units.

Another Example

Problem: The region between the line y = 2 and the curve y = x² (from x = 0 to x = √2) is rotated about the y-axis. Identify the axis of rotation and describe the resulting solid.
Step 1: The axis of rotation is the y-axis, which is the vertical line x = 0.
Axis of rotation: x=0\text{Axis of rotation: } x = 0
Step 2: As the region spins around the y-axis, each point at horizontal distance x from the y-axis traces a circle of radius x. The inner boundary is x = 0 and the outer boundary is x = √y (solving y = x² for x).
x=y,0≤y≤2x = \sqrt{y}, \quad 0 \le y \le 2
Step 3: The solid formed is a paraboloid-shaped bowl. You could find its volume using the disk method with respect to y.
V=π∫02(y)2 dy=π∫02y dy=2πV = \pi \int_0^2 (\sqrt{y})^2 \, dy = \pi \int_0^2 y \, dy = 2\pi
Answer: The axis of rotation is the y-axis (x = 0). The resulting solid of revolution is a paraboloid with volume 2π cubic units.

Frequently Asked Questions

Can the axis of rotation pass through the shape being rotated?
Yes. When the axis passes through or along the edge of the shape, you get a solid with no hole in the middle — like a cone or sphere. When the axis lies outside the shape (with a gap between them), the rotation produces a hollow solid like a torus (donut shape). The position of the axis relative to the region directly determines the geometry of the resulting solid.
How do you choose which axis to rotate around?
The axis of rotation is specified by the problem. Common choices are the x-axis, the y-axis, or a horizontal or vertical line like y = −1 or x = 5. Your choice of axis affects the shape and volume of the solid produced. Problems often ask you to compare the solids formed by rotating the same region about different axes.

Axis of Rotation vs. Axis of Symmetry

An axis of rotation is the line you physically spin a region around to create a 3D solid. An axis of symmetry is a line that divides a shape into two mirror-image halves. They are different concepts, though sometimes they coincide — for example, rotating a semicircle about its diameter (which is also its axis of symmetry) produces a sphere.

Why It Matters

The axis of rotation is central to calculus topics like computing volumes of solids of revolution using the disk, washer, and cylindrical shell methods. The choice of axis determines which integration method is most convenient and directly affects the setup of the integral. Beyond calculus, the concept appears in physics (angular momentum, torque) and engineering (lathe manufacturing, pottery wheels).

Common Mistakes

Mistake: Setting up the radius of rotation as the y-value of the curve when the axis of rotation is not the x-axis.
Correction: The radius is the distance from the curve to the axis of rotation, not simply the function value. If the axis is y = k, the radius is |f(x) − k|. Always measure the perpendicular distance from each point to the actual axis.
Mistake: Confusing rotation about the x-axis with rotation about the y-axis, leading to the wrong integral variable.
Correction: When rotating about the x-axis, cross-sectional disks are perpendicular to x, so you typically integrate with respect to x (or use shells with respect to y). When rotating about the y-axis, the roles reverse. Sketch the axis and the region before writing any integral.

Related Terms