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| Title of object: |
surface of revolution |
| Canonical Name: |
SurfaceOfRevolution |
| Type: |
Topic |
| Created on: |
2006-02-26 10:31:14 |
| Modified on: |
2006-02-26 11:35:34 |
| Classification: |
msc:45A05 |
| Synonyms: |
surface of revolution=area of revolution
surface of revolution=surface area of revolution |
Revision comment (for changes between this and next version):
| Changes for correction #7612 ('surface *area* of revolution'). |
Preamble:
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Content:
The \emph{surface of revolution} is a 3D surface, generated when an arc is rotated fully around a straight line.
The general surface of revolution is obtained when the arc is rotated about an axis. In Cartesian coordinates, the general surface of revolution A of a curve described by y in the interval [a, b] can be calculated by the formula
$$A = 2 \pi \int_{a}^{b} y \, \sqrt{ 1 + \left(\frac{dy}{dx}\right)^2 } \, dx$$
Similarly, rotated about the y-axis:
$$A = 2 \pi \int_{a}^{b} x \, \sqrt{ 1 + \left(\frac{dx}{dy}\right)^2 } \, dy$$
The general formula is most often seen with parametric coordinates. If x(t) and y(t) describe the curve, and x(t) is always positive or zero, then the general surface of revolution A in the interval [a, b] can be calulated by the formula
$$A = 2 \pi \int_{a}^{b} y \, \sqrt{ \left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2 } \, dt$$
To obtain a specific surface of revolution, translation or rotation can be used to move an arc before revolving it around an axis. For example, the specific surface of revolution around the line $y = s$ can be found by replacing y with $y - s$, moving the arc towards the x-axis so $y = s$ lies on it. Now, the surface of revolution can be found using one of the formulae above.
In this specific case, replacing y with $y = s$, the surface of revolution is found using the formula
$$A = 2 \pi \int_{a}^{b} (y-s) \sqrt{ \left(\frac{dy}{dx}\right)^2 } \, dy$$ |
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