Gaussian curvature of a surface of revolution

For the parametrization

X (x, θ) = (x, f (x) cos θ, f (x) sin θ),

the Gaussian curvature and area element are

K = - \frac{f ^{''}}{f ( 1 + f ^{'2} ) ^{2}}, d A = f 1 + f^{'2} d x d θ .

Table of contents
- Total Gaussian curvature of a surface-of-revolution strip Gaussian curvature of a surface of revolution

Total Gaussian curvature of a surface-of-revolution strip

For

a \leq x \leq b

, the curvature formula is an exact derivative and gives

\int K d A = - 2 π [\frac{f ^{'}}{1 + f ^{'2}}]_{a}^{b} .

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Past exam of the mathematics course of the University of Cambridge / 2024 / ib / Paper 4 / 11G / a / Solution

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