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Surfaces of revolution are 3D shapes created by rotating a 2D curve around an axis. This text delves into their geometry, calculus applications for surface area calculation, and practical uses in technology like parabolic antennas. It also discusses the computational challenges faced when determining these areas and the role of Computer Algebra Systems in solving complex integrals.

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## Definition

### Three-dimensional shapes

Surfaces of revolution are three-dimensional shapes formed by rotating a two-dimensional curve about a fixed axis

### Hollow with surface area

These surfaces are hollow, having no volume, yet possess a definable surface area

### Generation

To generate a surface of revolution, one takes a curve defined on the xy-plane and rotates it around a line known as the axis of revolution

## Calculus

### Surface area calculation

The surface area of a surface of revolution is calculated using integral calculus

### Formula

The formula for the surface area is \( S = 2\pi \int_a^b f(x)\sqrt{1+(f'(x))^2}\,dx \), where \( f(x) \) represents the generating curve, \( f'(x) \) is the derivative of \( f(x) \), and \( a \) and \( b \) define the interval over which the curve is rotated

### Derivation

The formula is derived by considering the surface area of a series of infinitesimally thin frustums created by the rotation of small segments of the curve

## Applications

### Practical uses

Surfaces of revolution have numerous practical applications, such as in telecommunications and in creating aesthetically pleasing shapes

### Examples

Examples include parabolic antennas and Gabriel's Horn, which showcase the surprising outcomes that can arise from infinite processes in calculus

### Surface area formula

The surface area formula is based on the geometry of a frustum and is used to calculate the surface area of these geometrically complex shapes

## Computation

### Mathematical challenges

The computation of the surface area can be mathematically challenging, requiring sophisticated techniques and the use of Computer Algebra Systems (CAS)

### Utility and beauty

Despite these difficulties, the study of surfaces of revolution exemplifies the utility and beauty of calculus in describing and solving complex problems