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Questions and Answers
What is the formula for computing the length of a closed piecewise smooth plane curve $\gamma : [a, b] \to \mathbb{R}^2$?
What is the formula for computing the length of a closed piecewise smooth plane curve $\gamma : [a, b] \to \mathbb{R}^2$?
- $L = \int_{a}^{b} \sqrt{x'(t)^{2} + y'(t)^{2}} dt$ (correct)
- $L = \int_{a}^{b} (x'(t) - y'(t)) dt$
- $L = \int_{a}^{b} (x'(t) + y'(t)) dt$
- $L = \int_{a}^{b} (x'(t) \cdot y'(t)) dt$
Who is the mathematician known to have approximated the perimeter of a circle by surrounding it with regular polygons?
Who is the mathematician known to have approximated the perimeter of a circle by surrounding it with regular polygons?
- Archimedes (correct)
- Pythagoras
- Euclid
- Newton
Which theory describes a generalized notion of perimeter, including hypersurfaces bounding volumes in n-dimensional Euclidean spaces?
Which theory describes a generalized notion of perimeter, including hypersurfaces bounding volumes in n-dimensional Euclidean spaces?
- Riemann integration
- Caccioppoli sets (correct)
- Complex analysis
- Fourier series
Why are polygons fundamental to determining perimeters?
Why are polygons fundamental to determining perimeters?
In the formula for computing the length of a closed piecewise smooth plane curve, what does $x'(t)$ represent?
In the formula for computing the length of a closed piecewise smooth plane curve, what does $x'(t)$ represent?
What is the measurement of space enclosed by a closed geometric figure?
What is the measurement of space enclosed by a closed geometric figure?
Which formula is used to calculate the total length of the boundary of a square field?
Which formula is used to calculate the total length of the boundary of a square field?
What is the formula for calculating the area of a triangle?
What is the formula for calculating the area of a triangle?
What is the formula for calculating the area of a circle?
What is the formula for calculating the area of a circle?
What is the formula for calculating the perimeter of a rectangle?
What is the formula for calculating the perimeter of a rectangle?
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