area of surface of revolutionSurfaceOfRevolution2006-02-26 10:31:142008-08-15 16:27:28Topic% this is the default PlanetMath preamble. as your knowledge
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% define commands hereA \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 arbitrary axis. If one chooses Cartesian coordinates, and specializes to the case of a surface of revolution generated by rotating about the x-axis a curve described by y in the interval $[a, b]$, its area can be calculated by the formula
$$A = 2 \pi \int_{a}^{b} y \, \sqrt{ 1 + \left(\frac{dy}{dx}\right)^2 } \, dx$$
Similarly, if the curve is rotated about the y-axis rather than the x-axis, one has the following formula:
$$A = 2 \pi \int_{a}^{b} x \, \sqrt{ 1 + \left(\frac{dx}{dy}\right)^2 } \, dy$$
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 area of a surface of revolution is found using the formula
$$A = 2 \pi \int_{a}^{b} (y-s) \sqrt{ \left(\frac{dy}{dx}\right)^2 } \, dy$$