Surface of revolution

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The parabola y = x2 rotated about the y-axis

The parabola y = x² rotated about the y-axis

A surface of revolution is a surface created by rotating a curve lying on some plane (the generatrix) around a straight line (the axis of rotation) that lies on the same plane.

Examples of surfaces generated by a straight line are the cylindrical and conical surfaces. A circle that is rotated about a (coplanar) axis through the center generates a sphere. If the axis is coplanar and outside the circle it generates a toroidal surface.

If the curve is described by the parametric functions $x (t)$ , $y (t)$ , with $t$ ranging over some interval $[a, b]$ , and the axis of revolution is the $y$ axis, then the area $A$ is given by the integral

$A = 2 \pi \int_a^b x(t) \ \sqrt{\left({dx \over dt}\right)^2 + \left({dy \over dt}\right)^2} \, dt,$

provided that $x (t)$ is never negative. This formula is the calculus equivalent of Pappus's centroid theorem. The quantity

$\left({dx \over dt}\right)^2 + \left({dy \over dt}\right)^2$

comes from the Pythagorean theorem and represents a small segment of the arc of the curve, as in the arc length formula. The quantity $2π x (t)$ is the path of (the centroid of) this small segment, as required by Pappus's theorem.

If the curve is described by the function y = f(x), a ≤ x ≤ b, then the integral becomes

$A=2\pi\int_a^b y \sqrt{1+\left(\frac{dy}{dx}\right)^2} \, dx$

for revolution around the x-axis, and

$A=2\pi\int_a^b x \sqrt{1+\left(\frac{dx}{dy}\right)^2} \, dy$

for revolution around the y-axis. These come from the above formula.

For example, the spherical surface with unit radius is generated by the curve x(t) = sin(t), y(t) = cos(t), when t ranges over $[0,π]$ . Its area is therefore

$A = 2 \pi \int_0^\pi \sin(t) \sqrt{\left(\cos(t)\right)^2 + \left(\sin(t)\right)^2} \, dt = 2 \pi \int_0^\pi \sin(t) \, dt = 4\pi.$

The use of surface of revolutions is essential in many fields in physics and engineering. When certain objects are designed digitally, revolutions like these can be used to determine surface area without the use of measuring the length and radius of the object being designed.

Solid of revolution
Gabriel's Horn
Canal surface - a generalisation of a surface of revolution

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Categories: Integral calculus | Surfaces

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Surface of revolution

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