Surface of Revolution
A surface of revolution is a surface in three-dimensional space created by rotating a curve, known as the generatrix, about a straight line in the same plane, known as the axis. In many cases, this axis is the x-axis or the y-axis.
Calculating Surface Area:
Revolution about the x-axis
For a curve defined by y = fx>0 on the interval a≤x≤b, the formula for the surface area is given by
Sx=2⁢π⁢∫abfx⁢1+f ′x2ⅆx = 2⁢π⁢∫aby⁢1+ⅆ yⅆ x2ⅆx
Revolution about the y-axis
For a curve defined by x = gy > 0 on the interval c≤y≤d, the formula for the surface area is given by
Sy=2⁢π⁢∫cdgy⁢1+g′y2ⅆy = 2⁢π⁢∫cdx⁢1+ⅆ xⅆ y 2ⅆy
For a curve defined parametrically by xt and yt:
The surface area obtained by rotating the curve around the x-axis for t ∈a,b is given by Sx=2⁢π⁢∫abyt⁢ⅆ xⅆ t2 + ⅆ yⅆ t2 ⅆt, provided that yt > 0 on this interval.
The surface area obtained by rotating the curve around the y-axis for t ∈ c,d is given by Sy=2⁢π⁢∫cdxt⁢ⅆ xⅆ t2 + ⅆ yⅆ t2 ⅆt, provided that xt > 0 on this interval.
Draw a curve in the plot on the left, choose an axis around which to rotate it, and click "Show Surface of Revolution" to view your surface of revolution and compute its surface area. Alternatively, you can select a predefined curve or enter a formula in the box.
Axis of Revolution:
Custom curvey = sin(x)y = xy = x^2y = x^3y = sqrt(x)y = exp(x)y = 5 fx=
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