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Questions and Answers
How is the length of a closed piecewise smooth plane curve $\gamma : [a, b] \to \mathbb{R}^2$ with $\gamma(t) = (x(t), y(t))$ computed?
How is the length of a closed piecewise smooth plane curve $\gamma : [a, b] \to \mathbb{R}^2$ with $\gamma(t) = (x(t), y(t))$ computed?
- $L = \int_{a}^{b}{x'(t)\,dt} + \int_{a}^{b}{y'(t)\,dt}$
- $L = \int_{a}^{b}{\sqrt{x'(t)^{2}+y'(t)^{2}}\,dt}$ (correct)
- $L = \int_{a}^{b}{(x'(t)+y'(t))\,dt}$
- $L = \int_{a}^{b}{\sqrt{x'(t)^{2}-y'(t)^{2}}\,dt}$
What theory describes the generalized notion of perimeter, including hypersurfaces bounding volumes in $n$-dimensional Euclidean spaces?
What theory describes the generalized notion of perimeter, including hypersurfaces bounding volumes in $n$-dimensional Euclidean spaces?
- Lebesgue integration
- Caccioppoli sets (correct)
- Fourier series
- Riemann sums
Who approximated the perimeter of a circle by surrounding it with regular polygons?
Who approximated the perimeter of a circle by surrounding it with regular polygons?
- Euclid
- Archimedes (correct)
- Pythagoras
- Newton
Which shapes are fundamental to determining perimeters because their perimeters are calculated by approximating them with sequences of polygons tending to these shapes?
Which shapes are fundamental to determining perimeters because their perimeters are calculated by approximating them with sequences of polygons tending to these shapes?
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What is the formula for calculating the length $L$ of a closed piecewise smooth plane curve $\gamma : [a, b] \to \mathbb{R}^2$ with $\gamma(t) = (x(t), y(t))$?
What is the formula for calculating the length $L$ of a closed piecewise smooth plane curve $\gamma : [a, b] \to \mathbb{R}^2$ with $\gamma(t) = (x(t), y(t))$?
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