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Curves, arc length and curvature

Parametrised curves, the unit tangent, curvature as the turning rate.

Arc length is ∫|r′(t)| dt; curvature is how fast the unit tangent turns per unit length. A circle of radius R has curvature 1/R. Picture it: the parabola y = x² curving most tightly at its vertex. Think it: curvature and torsion determine a space curve up to rigid motion (the fundamental theorem of curves).

Àwọn ààtò àwọn ìṣàmúlò-ètò · integrate sqrt(1 + 4x^2) dx from 0 to 1

Calculus · Definite-Integrals

Integrate sqrt(4x^2 + 1) from 0 to 1

\int_{0}^{1} \sqrt{4 x^{2} + 1}\, dx

Àwọn ìṣàmúlò-ètò

\int_{0}^{1} \sqrt{4 x^{2} + 1}\, dx
First find an antiderivative F, then evaluate F(b) − F(a).
\int \sqrt{4 x^{2} + 1}\, dx = \frac{x \sqrt{4 x^{2} + 1}}{2} + \frac{\operatorname{asinh}{\left(2 x \right)}}{4}
SqrtQuadratic rule.
F(1) - F(0) = \left(\frac{\operatorname{asinh}{\left(2 \right)}}{4} + \frac{\sqrt{5}}{2}\right) - \left(0\right)
Fundamental theorem of calculus: plug in the limits.
= \frac{\operatorname{asinh}{\left(2 \right)}}{4} + \frac{\sqrt{5}}{2} \approx 1.4789
Simplify.

Fi àwọn àgbèwọlé hàn

\frac{\operatorname{asinh}{\left(2 \right)}}{4} + \frac{\sqrt{5}}{2}

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Bii o ṣe le: Curves, arc length and curvature

First find an antiderivative F, then evaluate F(b) − F(a).
SqrtQuadratic rule.
Fundamental theorem of calculus: plug in the limits.
Simplify.

Àwọn Àtòjọ-ẹ̀yàn

What is curvature?

For a curve, how fast its direction turns per unit length; for a surface, Gauss's combination of the two principal bendings, and Gauss's theorem says it can be measured from inside the surface without leaving it.

Diẹ̀ nínú Differential Geometry

Surfaces and Gaussian curvature