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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).
İşlənmiş nümunə · integrate sqrt(1 + 4x^2) dx from 0 to 1
Integrate sqrt(4x^2 + 1) from 0 to 1
Addım-addım
- \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.
Cavabı göstər
İndi siz Problem seçin, ya da özünü yazın ya da çəkin. Hər addım, bir rəsm, cavab siz soruşana qədər gizli qalır.
Burada istifadə edilən simvollar
Tam təsviri, şəkli və hər hərfin nə demək olduğunu görmək üçün hər hansı simvolu vurun.
Növbəti: 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.
İnsanlar soruşurlar
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.