Revision #2795 → #3275 · back to history
modifiedArc length as integral of speed660fc93d46c2
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| anchors | [{"section":"As an integral","snippet":"the arc length is obtained by integrating speed"},{"type":"math_alttext","value":"{\\displaystyle L=\\int _{a}^{b}{\\sqrt {x'(t)^{2}+y'(t)^{2}}}\\,dt.}"}] | — |
modifiedArc length formula for differentiable curve851e801fce40
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| anchors | [{"section":"As an integral","snippet":"The length of the curve is given by the formula"},{"type":"math_alttext","value":"{\\displaystyle L(f)=\\int _{a}^{b}|f'(t)|\\,dt}"}] | — |
modifiedArc length for explicit function y=f(x)a305d09d31a7
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| anchors | [{"section":"Finding arc lengths by integration","snippet":"then it is simply a special case of a parametric equation"},{"type":"math_alttext","value":"{\\displaystyle {\\sqrt {dx^{2}+dy^{2}}}={\\sqrt {1+\\left({\\frac {dy}{dx}}\\right)^{2}\\,}}dx.}"}] | — |
modifiedArc length in polar coordinatesd3d0f5574b1d
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| anchors | [{"section":"Other coordinate systems","snippet":"So for a curve expressed in polar coordinates, the arc length is"},{"type":"math_alttext","value":"{\\displaystyle \\int _{t_{1}}^{t_{2}}{\\sqrt {\\left({\\frac {dr}{dt}}\\right)^{2}+r^{2}\\left({\\frac {d\\theta }{dt}}\\right)^{2}\\,}}dt=\\int _{\\theta (t_{1})}^{\\theta (t_{2})}{\\sqrt {\\left({\\frac {dr}{d\\theta }}\\right)^{2}+r^{2}\\,}}d\\theta .}"}] | — |
modifiedArc length in spherical coordinates4eb3c41774a7
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| anchors | [{"section":"Other coordinate systems","snippet":"So for a curve expressed in spherical coordinates, the arc length is"},{"type":"math_alttext","value":"{\\displaystyle \\int _{t_{1}}^{t_{2}}{\\sqrt {\\left({\\frac {dr}{dt}}\\right)^{2}+r^{2}\\left({\\frac {d\\theta }{dt}}\\right)^{2}+r^{2}\\sin ^{2}\\theta \\left({\\frac {d\\phi }{dt}}\\right)^{2}\\,}}dt.}"}] | — |
modifiedArc length in cylindrical coordinatesc595dbeaa873
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| anchors | [{"section":"Other coordinate systems","snippet":"the arc length of a curve expressed in cylindrical coordinates is"},{"type":"math_alttext","value":"{\\displaystyle \\int _{t_{1}}^{t_{2}}{\\sqrt {\\left({\\frac {dr}{dt}}\\right)^{2}+r^{2}\\left({\\frac {d\\theta }{dt}}\\right)^{2}+\\left({\\frac {dz}{dt}}\\right)^{2}\\,}}dt.}"}] | — |
addedArc length as distance along a curvef2dc35871fed
addedHausdorff dimension/measure for non-rectifiable curvesf67e35eb548d