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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).

କାର୍ଯ୍ୟକାରୀ ଉଦାହରଣ: 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

ପଦକ୍ଷେପ କ୍ରମେ

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

    First find an antiderivative F, then evaluate F(b) − F(a).

  2. \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.

  3. 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.

  4. = \frac{\operatorname{asinh}{\left(2 \right)}}{4} + \frac{\sqrt{5}}{2} \approx 1.4789

    Simplify.

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

Symbols used here

\int f(x)\,dx,\ \int_a^b
integral
Antiderivative (indefinite) or signed area from a to b (definite).
\sqrt{x},\ \sqrt[n]{x}
square root, n-th root
The non-negative number whose square (n-th power) is x.
\approx
approximately equal
Equal to the precision shown, not exactly.
\leq,\ \geq
less/greater than or equal
Inequalities that allow equality; < and > exclude it.
f'(x),\ \frac{dy}{dx}
derivative
Instantaneous rate of change; slope of the graph.
\frac{\partial f}{\partial x}
partial derivative
Derivative with respect to x, holding the other variables fixed.
\nabla f
gradient (nabla, del)
Vector of partial derivatives; points uphill.
\kappa,\ K
curvature, Gaussian curvature
How fast a curve turns; the product of a surface's principal curvatures.

How to: Curves, arc length and curvature

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

Questions people ask

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.

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ଅଧିକ Differential Geometry

Surfaces and Gaussian curvature