CalculusLesson 4 of 5 in the course Applications of integration

Surfaces of Revolution: Geometry, Area, Minimal Surfaces, and Geodesics

5 flashcards · 5 quiz questions · 5 match pairs · 5 fill-in sentences · English

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Summary

A surface of revolution is formed by rotating a curve around an axis. This lesson develops the geometric vocabulary and shows how to describe these surfaces with cross sections and coordinates.

The generatrix is the curve being rotated, and the axis of rotation is the fixed line around which it turns. A full revolution sweeps the curve into a two-dimensional surface. The volume enclosed by that surface is called a solid of revolution.

All flashcards

What is a surface of revolution?
A surface formed when a curve, called the generatrix, makes one full rotation around a fixed axis.
How do meridional sections differ from perpendicular sections?
A meridional section is cut by a plane containing the axis and shows the generating profile. A perpendicular section is cut by a plane perpendicular to the axis and is circular, apart from possible degenerate endpoint cases.
What does rotating a circle around one of its diameters produce?
It produces a sphere, and the original circle becomes a great circle of that sphere.
How can a profile y=f(x)y=f(x) be represented after rotation around the xx-axis?
Its points satisfy y2+z2=f(x)2y^2+z^2=f(x)^2, and a parametrization is (x,f(x)cos⁡θ,f(x)sin⁡θ)(x,f(x)\cos\theta,f(x)\sin\theta) for 0≤θ≤2π0\leq\theta\leq2\pi.
What is a toroid, and how does a torus relate to it?
A toroid is a surface of revolution with a hole because the axis does not intersect the surface. A torus is the special case generated by revolving a circle.

Quiz questions

  1. What is the curve being rotated to create a surface of revolution called?Answer: The generatrix
  2. What shape is generated when a straight line parallel to, but not coincident with, the axis of rotation is revolved?Answer: A cylindrical surface
  3. What is a meridional section of a surface of revolution?Answer: A section made by a plane that contains the axis of rotation
  4. A circle of radius rr is revolved around an axis in its plane, at a distance greater than rr from its center. What surface is generated?Answer: A ring-shaped torus
  5. Which parametrization describes rotation of the plane curve (x(t),y(t))(x(t),y(t)) around the xx-axis?Answer: (x(t),y(t)cos⁡θ,y(t)sin⁡θ)(x(t),y(t)\cos\theta,y(t)\sin\theta)

Based on the Wikipedia article Surface of revolution, licensed CC BY-SA 4.0. Adapted into a summary, flashcards and quizzes.

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