--- title: "NLineIntegrate" language: "en" type: "Symbol" summary: "NLineIntegrate[f, {x, y, ...} \\[Element] curve] computes the numerical scalar line integral of the function f[x, y, ...] over the curve. NLineIntegrate[{p, q, ...}, {x, y, ...} \\[Element] curve] computes the numerical vector line integral of the vector function {p[x, y, ...], q[x, y, ...], ...}." keywords: - line integral - path integral - vector calculus - Green's theorem - contour integral - work of a force - vector field - potential - scalar field - moment of inertia - centroid - area - college calculus - path dependence - conservative force - non-conservative force - numerical integral canonical_url: "https://reference.wolfram.com/language/ref/NLineIntegrate.html" source: "Wolfram Language Documentation" related_guides: - title: "Vector Analysis" link: "https://reference.wolfram.com/language/guide/VectorAnalysis.en.md" - title: "Symbolic Vectors, Matrices and Arrays" link: "https://reference.wolfram.com/language/guide/SymbolicArrays.en.md" related_functions: - title: "LineIntegrate" link: "https://reference.wolfram.com/language/ref/LineIntegrate.en.md" - title: "SurfaceIntegrate" link: "https://reference.wolfram.com/language/ref/SurfaceIntegrate.en.md" - title: "NSurfaceIntegrate" link: "https://reference.wolfram.com/language/ref/NSurfaceIntegrate.en.md" - title: "ContourIntegrate" link: "https://reference.wolfram.com/language/ref/ContourIntegrate.en.md" - title: "NContourIntegrate" link: "https://reference.wolfram.com/language/ref/NContourIntegrate.en.md" - title: "Integrate" link: "https://reference.wolfram.com/language/ref/Integrate.en.md" - title: "NIntegrate" link: "https://reference.wolfram.com/language/ref/NIntegrate.en.md" - title: "Region" link: "https://reference.wolfram.com/language/ref/Region.en.md" - title: "ParametricRegion" link: "https://reference.wolfram.com/language/ref/ParametricRegion.en.md" - title: "ArcLength" link: "https://reference.wolfram.com/language/ref/ArcLength.en.md" - title: "Grad" link: "https://reference.wolfram.com/language/ref/Grad.en.md" --- [EXPERIMENTAL] # NLineIntegrate NLineIntegrate[f, {x, y, …}∈curve] computes the numerical scalar line integral of the function f[x, y, …] over the curve. NLineIntegrate[{p, q, …}, {x, y, …}∈curve] computes the numerical vector line integral of the vector function {p[x, y, …], q[x, y, …], …}. ## Details and Options * Line integrals are also known as curve integrals and work integrals. * Scalar line integrals integrate scalar functions along a curve. They typically compute things like length, mass and charge for a curve. * Vector line integrals are used to compute the work done by a vector function along a curve in the direction of its tangent. Typical vector functions include a force field, electric field and fluid velocity field. * The scalar line integral of the function ``f`` along a ``curve`` $C=\left\{r(u)\in \mathbb{R}^n,a\leq u\leq b\right\}$ is given by: [image] * … where $$\left\| r'(u)\right\|$$ is the measure of a parametric curve segment. * The scalar line integral is independent of the parametrization and orientation of the ``curve``. Any one-dimensional ``RegionQ`` object can be used as a ``curve``. * The vector line integral of the function ``F`` along a ``curve`` $C=\left\{r(u)\in \mathbb{R}^n,a\leq u\leq b\right\}$ is given by: [image] * … where $$F(r(u)).r'(u)$$ is projection of the vector function $F$ onto the tangent direction so only the component in the tangent direction gets integrated. * The vector line integral is independent of the parametrization of the curve, but it does depend on the orientation of the curve. * The orientation for a curve is given by a tangent vector field $T(u)$ over the curve. [image] * For a parametric curve ``ParametricRegion[{r1[u], …, rn[u]}, …]``, the tangent vector field $T(u)$ is taken to be ``∂ur[u]``. * Special curves in $\mathbb{R}^2$ with their assumed tangent orientations include: | | | | | ------- | ------------------------------------------ | --------------------------------------------------------------------------------- | | [image] | Line[{p1, p2, …}] | the orientation follows the points in the order they are given from p1 to p2 etc. | | [image] | HalfLine[{p1, p2}] HalfLine[p, v] | the orientation is from p1 to p2 or in the v direction | | [image] | InfiniteLine[{p1, p2}] InfiniteLine[p, v] | the orientation is from p1 to p2 or in the v direction | | [image] | Circle[p, r] | the orientation is counterclockwise | * Special curves in $\mathbb{R}^3$ with their assumed tangent orientations include: | | | | | ------- | ------------------------------------------ | --------------------------------------------------------------------------------- | | [image] | Line[{p1, p2, …}] | the orientation follows the points in the order they are given from p1 to p2 etc. | | [image] | HalfLine[{p1, p2}] HalfLine[p, v] | the orientation is from p1 to p2 or in the v direction | | [image] | InfiniteLine[{p1, p2}] InfiniteLine[p, v] | the orientation is from p1 to p2 or in the v direction | * Special curves in $\mathbb{R}^n$ with their assumed tangent orientations include: | | | | | ------- | ------------------------------------------ | -------------------------------------------------------------- | | [image] | Line[{p1, p2, …}] | the orientation follows the points in the order they are given | | [image] | HalfLine[{p1, p2}] HalfLine[p, v] | the orientation is from p1 to p2 the orientation is given by v | | [image] | InfiniteLine[{p1, p2}] InfiniteLine[p, v] | the orientation is from p1 to p2 the orientation is given by v | * The coordinates along the ``curve`` can be specified using ``VectorSymbol``. » * The following options can be given: | | | | | ----------------- | ---------------- | ------------------------------------------- | | AccuracyGoal | Infinity | digits of absolute accuracy sought | | MaxPoints | Automatic | maximum total number of sample points | | MaxRecursion | Automatic | maximum number of recursive subdivisions | | Method | Automatic | method to use | | MinRecursion | 0 | minimum number of recursive subdivisions | | PrecisionGoal | Automatic | digits of precision sought | | WorkingPrecision | MachinePrecision | the precision used in internal computations | --- ## Examples (82) ### Basic Examples (7) Line integral of a scalar field over a circle: ```wl In[1]:= NLineIntegrate[2x, {x, y}∈Circle[{0, 0}, 1, {0, Pi / 2}]] Out[1]= 2. ``` --- Line integral of a vector field over a line segment: ```wl In[1]:= NLineIntegrate[{x * y, x - y}, {x, y}∈Line[{{0, 0}, {1, 1}}]] Out[1]= 0.333333 ``` --- Line integral of a vector field over a space curve: ```wl In[1]:= NLineIntegrate[{x ^ 2 y z, 3x y, y ^ 2}, {x, y, z}∈ParametricRegion[{Cos[t], Sin[t], t}, {{t, 0, Pi / 2}}]] Out[1]= 1.63119 ``` --- Line integral of a scalar field in two dimensions: ```wl In[1]:= f = y ^ 2; ``` Curve over which to integrate: ```wl In[2]:= reg = ParametricRegion[{{t ^ (5 / 2), t}, 0 <= t <= 2}, {t}]; ``` A contour plot of $f$ and the curve: ```wl In[3]:= Show[ContourPlot[f, {x, 0, 8}, {y, 0, 2}], ...] Out[3]= [image] ``` The line integral: ```wl In[4]:= NLineIntegrate[f, {x, y}∈reg] Out[4]= 12.9142 ``` --- Line integral of a vector field in two dimensions: ```wl In[1]:= f = {x + 2y, x ^ 2}; In[2]:= reg = Line[{{0, 0}, {2, 3 / 2}, {3, 0}}]; ``` The vector field and the integration path: ```wl In[3]:= VectorPlot[f, {x, 0, 3}, {y, 0, 2}, ...] Out[3]= [image] ``` The line integral: ```wl In[4]:= NLineIntegrate[f, {x, y}∈reg] Out[4]= 1.5 ``` --- Line integral of a vector field in three dimensions: ```wl In[1]:= f = {z ^ 3, x ^ 3, y ^ 3}; In[2]:= reg = Line[{{1, 1 / 2, 0}, {3, 2, 2}}]; In[3]:= Show[VectorPlot3D[f, {x, 0, 4}, {y, 0, 2}, {z, 0, 2}, Rule[...]], ...] Out[3]= [image] In[4]:= NLineIntegrate[f, {x, y, z}∈reg] Out[4]= 24.3125 ``` --- Use ``VectorSymbol`` : ```wl In[1]:= NLineIntegrate[VectorSymbol["x"].VectorSymbol["x"], VectorSymbol["x"]∈Line[{{0, 0}, {0, 1}}]] Out[1]= 0.333333 ``` Use implicit vector notation: ```wl In[2]:= NLineIntegrate[x, x∈Line[{{0, 0}, {0, 1}}]] Out[2]= 0.5 ``` ### Scope (33) #### Basic Uses (4) Line integral of a scalar field: ```wl In[1]:= f = Exp[x]; In[2]:= NLineIntegrate[f, {x, y}∈Line[{{0, 0}, {2, 3}}]] Out[2]= 11.518 ``` --- Line integral of a vector field in three dimensions: ```wl In[1]:= f = {x, x - y, z x ^ 2}; In[2]:= NLineIntegrate[f, {x, y, z}∈ParametricRegion[{Sin[t], t, 2t}, {{t, 0, Pi}}]] Out[2]= 6.9348 ``` --- ``LineIntegrate`` works with many special curves: ```wl In[1]:= f = {x ^ 2 y, x}; In[2]:= NLineIntegrate[f, {x, y}∈Annulus[{0, 0}, {1, 2}, {0, Pi / 4}]]//Quiet Out[2]= -1.23202 ``` --- Line integral over a parametric curve: ```wl In[1]:= f = {Log[x], y}; In[2]:= reg = ParametricRegion[{2t + 1, t ^ 2}, {{t, 0, 1}}]; In[3]:= NLineIntegrate[f, {x, y}∈reg] Out[3]= 1.79584 ``` #### Scalar Functions (11) Line integral of a scalar field over a curve: ```wl In[1]:= f = 2Abs[x - y] ^ (3 / 4); In[2]:= reg = ParametricRegion[{t, t ^ (3 / 2)}, {{t, 0, 8}}]; ``` Contour plot of $f$ and the curve: ```wl In[3]:= Show[ContourPlot[f, {x, 0, 8}, {y, 0, 25}], ...] Out[3]= [image] ``` Line integral: ```wl In[4]:= NLineIntegrate[f, {x, y}∈reg] Out[4]= 181.377 ``` --- Line integral of a scalar field over an arc of a circle: ```wl In[1]:= f = 1 / (x + Sin[y] + 1) ^ 3; In[2]:= reg = Circle[{0, 0}, 1, {0, Pi / 2}]; In[3]:= ContourPlot[f, {x, 0, 1}, {y, 0, 1}, Epilog -> reg, ImageSize -> Small] Out[3]= [image] In[4]:= NLineIntegrate[f, {x, y}∈reg] Out[4]= 0.150608 ``` --- Line integral of a scalar field over a parametric curve: ```wl In[1]:= f = 1 / (x + 1) - y / 5; In[2]:= reg = ParametricRegion[{3 / 2t ^ 2, 4t}, {{t, -1, 1}}]; ``` Contour plot of the function and the curve: ```wl In[3]:= Show[ContourPlot[f, {x, -1 / 2, 2}, {y, -4, 4}], ...] Out[3]= [image] In[4]:= NLineIntegrate[f, {x, y}∈reg] Out[4]= 6.17595 ``` --- Line integral of a scalar field over a circle: ```wl In[1]:= f = Sin[1 / (x ^ 3 + y + 1)]; In[2]:= reg = Circle[{2, 3}, 3]; In[3]:= ContourPlot[f, {x, -1, 5}, {y, 0, 6}, Epilog -> reg, ImageSize -> Small] Out[3]= [image] In[4]:= NLineIntegrate[f, {x, y}∈reg] Out[4]= 3.00069 ``` --- Line integral of a scalar field over a space curve: ```wl In[1]:= f = x + y - z; In[2]:= reg = ParametricRegion[{t ^ 3 / 3, Sqrt[2]t ^ 2 / 2, t}, {{t, 0, 1}}]; ``` Line integral: ```wl In[3]:= NLineIntegrate[f, {x, y, z}∈reg] Out[3]= -0.233987 ``` --- Line integral of a scalar field over the boundary of an annulus: ```wl In[1]:= f = 1 / Exp[x + y + 5]; In[2]:= reg = Annulus[{0, 0}, {1, 2}, {0, 3 / 4Pi}]; ``` Contour plot of the function and the curve: ```wl In[3]:= ContourPlot[f, {x, -2, 4}, {y, -1, 5}, ...] Out[3]= [image] In[4]:= NLineIntegrate[f, {x, y}∈reg] Out[4]= 0.0206253 ``` --- Line integral of a scalar field over a closed polygon: ```wl In[1]:= f = Exp[Cos[Abs[x / 2 - y]]]; In[2]:= reg = Line[{{0, 0}, {1, 2}, {0, 1}, {0, 0}}]; ``` Contour plot of the function and the curve: ```wl In[3]:= ContourPlot[f, {x, 0, 1}, {y, 0, 2}, Epilog -> reg, AspectRatio -> Automatic] Out[3]= [image] In[4]:= NLineIntegrate[f, {x, y}∈reg] Out[4]= 8.80999 ``` --- Line integral of a scalar field over an elliptical path: ```wl In[1]:= f = 2 x y; In[2]:= reg = Circle[{0, 0}, {1, 1 / Sqrt[2]}]; ``` Contour plot of the function and the curve: ```wl In[3]:= ContourPlot[f, {x, -1, 1}, {y, -1, 1}, Epilog -> reg, ImageSize -> Small] Out[3]= [image] ``` Line integral: ```wl In[4]:= NLineIntegrate[f, {x, y}∈reg]//Quiet Out[4]= -5.551115123125783`*^-17 ``` --- Line integral of a scalar field over a parametric curve: ```wl In[1]:= f = x / 2 + Sin[y] ^ 2; In[2]:= reg = ParametricRegion[{6Cos[t] - Cos[6t], 6Sin[t] - Sin[6t]}, {{t, 0, 2Pi}}]; In[3]:= Show[ContourPlot[f, {x, -10, 10}, {y, -10, 10}], RegionPlot[reg, BoundaryStyle -> Red], ImageSize -> Small] Out[3]= [image] ``` Line integral: ```wl In[4]:= NLineIntegrate[f, {x, y}∈reg, MaxRecursion -> 20] Out[4]= 21.1214 ``` --- Line integral of a scalar field over a circle: ```wl In[1]:= f = ArcTan[(y / x) ^ 2]; In[2]:= reg = Circle[{0, 0}, 4]; ``` Contour plot of $f$ and the curve: ```wl In[3]:= ContourPlot[f, {x, -4, 4}, {y, -4, 4}, Epilog -> reg, ImageSize -> Small] Out[3]= [image] ``` Line integral: ```wl In[4]:= NLineIntegrate[f, {x, y}∈reg] Out[4]= 19.7392 ``` --- Line integral of a scalar field over the boundary of a sector of a disk: ```wl In[1]:= f = Cos[x - y]; In[2]:= reg = Disk[{0, 0}, 3, {0, 3Pi / 4}]; ``` Contour plot of $f$ and the curve: ```wl In[3]:= ContourPlot[f, {x, -3, 3}, {y, -1, 5}, ...] Out[3]= [image] ``` Line integral: ```wl In[4]:= NLineIntegrate[f, {x, y}∈reg] Out[4]= -2.27712 ``` #### Vector Functions (12) Line integral of a vector field in three dimensions over a parametrized curve: ```wl In[1]:= f = {2x, x, 1}; In[2]:= reg = ParametricRegion[{Sin[t], Cos[t], t}, {{t, 0, 2Pi}}]; ``` Visualization of the vector field and the curve: ```wl In[3]:= Show[VectorPlot3D[f, {x, -1, 1}, {y, -1, 1}, {z, 0, 7}, ...], ...] Out[3]= [image] ``` Line integral: ```wl In[4]:= NLineIntegrate[f, {x, y, z}∈reg] Out[4]= 3.14159 ``` --- Line integral of a vector field over a curve in two dimensions: ```wl In[1]:= f = {Sin[x], y}; In[2]:= reg = ParametricRegion[{t ^ 2, t ^ 3}, {{t, 0, 1}}]; In[3]:= Show[VectorPlot[f, {x, 0, 1}, {y, 0, 1}, Rule[...]], ...] Out[3]= [image] ``` Line integral: ```wl In[4]:= NLineIntegrate[f, {x, y}∈reg] Out[4]= 0.959698 ``` --- Line integral of a vector field over a circular arc: ```wl In[1]:= f = {x * (x ^ 2 + y ^ 2), y * (x ^ 2 + y ^ 2)}; In[2]:= reg = Circle[{0, 0}, 1, {0, Pi / 2}]; In[3]:= VectorPlot[f, {x, 0, 1}, {y, 0, 1}, ...] Out[3]= [image] In[4]:= NLineIntegrate[f, {x, y}∈reg]//Quiet Out[4]= 0. ``` --- Line integral of a vector field over a line segment: ```wl In[1]:= f = {Sqrt[x ^ 2 + y ^ 2], Sqrt[Abs[x + y]]}; In[2]:= reg = Line[{{0, 0}, {2, 2}, {4, 0}}]; In[3]:= VectorPlot[f, {x, -1, 5}, {y, -1, 3}, ...] Out[3]= [image] In[4]:= NLineIntegrate[f, {x, y}∈reg] Out[4]= 7.98799 ``` --- Line integral of a vector field over a parametrized curve in three dimensions: ```wl In[1]:= f = {x, x ^ 2 y ^ 2, 0}; In[2]:= reg = ParametricRegion[{Cos[t] ^ 2, Cos[t]Sin[t], Sqrt[Cos[t](1 - Cos[t])]}, {{t, 0, 2Pi}}]; In[3]:= Show[VectorPlot3D[f, {x, -1, 1}, {y, -1, 1}, {z, 0, 1}, Rule[...]], ...] Out[3]= [image] In[4]:= NLineIntegrate[f, {x, y, z}∈reg] Out[4]= 0.0981748 ``` --- Line integral of a vector field over a curve: ```wl In[1]:= f = {x + y, y + z, z + x}; In[2]:= reg = ParametricRegion[{Sin[t] ^ 2 / Sqrt[2] + Cos[t] ^ 2, (1 - 1 / Sqrt[2])Cos[t]Sin[t], Sin[t] / Sqrt[2]}, {{t, 0, 2Pi}}]; In[3]:= Show[VectorPlot3D[f, {x, 0, 1}, {y, -1, 1}, {z, -1, 1}, ...], ...] Out[3]= [image] In[4]:= NLineIntegrate[f, {x, y, z}∈reg] Out[4]= -0.134753 ``` --- Line integral of a vector field over an elliptical arc: ```wl In[1]:= f = {x, Sin[x y ^ 2]}; In[2]:= reg = Circle[{0, 0}, {2, 1}, {0, Pi}]; In[3]:= VectorPlot[f, {x, -2, 2}, {y, -1 / 2, 3 / 2}, ...] Out[3]= [image] In[4]:= NLineIntegrate[f, {x, y}∈reg] Out[4]= 0.737448 ``` --- Line integral of a vector field over a parametric curve: ```wl In[1]:= f = {x ^ 2, x ^ 2, Exp[y ^ 2 / 4]}; In[2]:= reg = ParametricRegion[{(2 + Cos[3t])Cos[t], (2 + Cos[3t])Sin[t], Sin[3t]}, {{t, 0, 2Pi}}]; In[3]:= Show[VectorPlot3D[f, {x, -3, 3}, {y, -3, 3}, {z, -1, 1}, ...], ...] Out[3]= [image] In[4]:= NLineIntegrate[f, {x, y, z}∈reg] Out[4]= 12.6203 ``` --- Line integral of a vector field over a parametric curve in three dimensions: ```wl In[1]:= f = {(y - 1) ^ 2z, x ^ 2y, 2x y}; In[2]:= reg = ParametricRegion[{Cos[t], Sin[t] / 2, Sin[t] / 2}, {{t, 0, Pi}}]; In[3]:= Show[VectorPlot3D[f, {x, -1, 1}, {y, -1, 1}, {z, -1, 1}, Rule[...]], ...] Out[3]= [image] In[4]:= NLineIntegrate[f, {x, y, z}∈reg] Out[4]= 0.0673397 ``` --- Line integral of a vector field over a parametrized curve: ```wl In[1]:= f = {x ^ 2, 1 / 3, z ^ 2}; In[2]:= reg = ParametricRegion[{2 / (1 + t ^ 2), 3t, 2t / (1 + t ^ 2)}, {{t, -10, 10}}]; In[3]:= Show[VectorPlot3D[f, {x, 0, 2}, {y, -30, 30}, {z, -1, 1}, ...], ...] Out[3]= [image] In[4]:= NLineIntegrate[f, {x, y, z}∈reg] Out[4]= 20.0052 ``` --- Line integral of a vector field over an elliptical path: ```wl In[1]:= f = {Sin[x ^ 4 y ^ 2], Sin[y]}; In[2]:= reg = Circle[{0, 1}, {Sqrt[2], 1}]; In[3]:= VectorPlot[f, {x, -2, 2}, {y, 0, 2}, ...] Out[3]= [image] In[4]:= NLineIntegrate[f, {x, y}∈reg] Out[4]= 0.250465 ``` --- Line integral of a vector field in higher dimensions: ```wl In[1]:= f = {x y, x ^ 2t z, y t, z t}; In[2]:= NLineIntegrate[f, {x, y, z, t}∈Line[{{0, 0, 0, 0}, {1, 2, 3, 4}}]] Out[2]= 29.4667 ``` #### Special Curves (4) Line integral over a circular arc: ```wl In[1]:= f = Sqrt[x] * Abs[y] ^ (3 / 4); In[2]:= reg = Circle[{0, 0}, 4, {-Pi / 2, Pi / 2}]; In[3]:= ContourPlot[f, {x, 0, 4}, {y, -4, 4}, ...] Out[3]= [image] In[4]:= NLineIntegrate[f, {x, y}∈reg] Out[4]= 33.6993 ``` --- Line integral of a vector field over the boundary of a circular sector of radius 1: ```wl In[1]:= f = {Exp[Sin[3y]], y + x ^ 2}; In[2]:= reg = Disk[{0, 0}, 1, {0, Pi / 2}]; In[3]:= VectorPlot[f, {x, 0, 1}, {y, 0, 1}, ...] Out[3]= [image] In[4]:= NLineIntegrate[f, {x, y}∈reg] Out[4]= -0.117371 ``` --- Line integral over a polygon: ```wl In[1]:= f = {1 / (x + 2y) ^ 2, -1 / (x + y)}; In[2]:= reg = Line[{{1, 0}, {0, 1}, {-1, 0}, {0, -1}, {1, 0}}]; In[3]:= VectorPlot[f, {x, -1, 1}, {y, -1, 1}, ...] Out[3]= [image] In[4]:= NLineIntegrate[f, {x, y}∈reg]//Quiet Out[4]= -3.38629 ``` --- Line integral over the boundary of an annulus: ```wl In[1]:= f = {-2y, x}; In[2]:= reg = Annulus[{0, 0}, {1, 2}, {0, Pi / 4}]; In[3]:= VectorPlot[f, {x, 0, 2}, {y, 0, 2}, ...] Out[3]= [image] In[4]:= NLineIntegrate[f, {x, y}∈reg, AccuracyGoal -> 5] Out[4]= 3.53429 ``` #### Parametric Curves (2) Line integral of a vector field over a spiral in three dimensions: ```wl In[1]:= f = {y - z, z - x, x - y}; In[2]:= reg = ParametricRegion[{ Cos[t], Sin[t], t}, {{t, 0, 2Pi}}]; In[3]:= Show[VectorPlot3D[f, {x, -1, 1}, {y, -1, 1}, {z, 0, 2Pi}, ...], ...] Out[3]= [image] In[4]:= NLineIntegrate[f, {x, y, z}∈reg] Out[4]= -12.5664 ``` --- Line integral of a scalar field over a parametric curve: ```wl In[1]:= NLineIntegrate[Exp[-x], {x, y}∈ParametricRegion[{2t, 3t}, {{t, 0, Infinity}}]] Out[1]= 1.80278 ``` ### Options (8) #### AccuracyGoal (1) The option ``AccuracyGoal`` sets the number of digits of accuracy: ```wl In[1]:= exact = LineIntegrate[{1, x ^ 2}, {x, y}∈Circle[]] Out[1]= 0 In[2]:= NLineIntegrate[{1, x ^ 2}, {x, y}∈Circle[], AccuracyGoal -> 15] - exact Out[2]= -7.355227538141662`*^-16 ``` The result with default settings only sets a ``PrecisionGoal`` : ```wl In[3]:= NLineIntegrate[{1, x ^ 2}, {x, y}∈Circle[]] - exact ``` NIntegrate::ncvb: NIntegrate failed to converge to prescribed accuracy after 9 recursive bisections in \[FormalT]\$239468 near {\[FormalT]\$239468} = {2.88379}. NIntegrate obtained -7.66748\*10^-16 and 6.36385740047725\`\*^-16 for the integral and error estimates. ```wl Out[3]= -7.667477763817487`*^-16 ``` #### MaxPoints (1) The option ``MaxPoints`` stops the integration after a specified number of points has been evaluated: ```wl In[1]:= NLineIntegrate[1 / Sqrt[x ^ 4 + y ^ 4], {x, y}∈Circle[], MaxPoints -> 10] ``` NIntegrate::maxp: The integral failed to converge after 33 integrand evaluations. NIntegrate obtained 7.416018317455614\` and 0.11469735508611181\` for the integral and error estimates. ```wl Out[1]= 7.41602 ``` With default options: ```wl In[2]:= NLineIntegrate[1 / Sqrt[x ^ 4 + y ^ 4], {x, y}∈Circle[]] Out[2]= 7.4163 ``` #### MaxRecursion (1) The option ``MaxRecursion`` specifies the maximum number of recursive steps: ```wl In[1]:= reg = ParametricRegion[{Sin[t], 1}, {{t, 0, Pi}}]; In[2]:= NLineIntegrate[{1 / Sqrt[x], x}, {x, y}∈reg] ``` NIntegrate::ncvb: NIntegrate failed to converge to prescribed accuracy after 9 recursive bisections in t near {t} = {3.14159265358978716777886064424740007652546292834661175678162559066}. NIntegrate obtained 2.2125866659672422\`\*^-8 and 1.0493182805081925\`\*^-9 for the integral and error estimates. ```wl Out[2]= 2.2125866659672422`*^-8 ``` Increasing the number of recursions: ```wl In[3]:= NLineIntegrate[{1 / Sqrt[x], x}, {x, y}∈reg, MaxRecursion -> 20] Out[3]= 2.2132751187776273`*^-8 ``` The exact result is: ```wl In[4]:= LineIntegrate[{1 / Sqrt[x], x}, {x, y}∈reg] Out[4]= 0 ``` #### Method (1) The option ``Method`` can take the same values as in ``NIntegrate``. For example: ```wl In[1]:= NLineIntegrate[{-Sin[y] ^ 3, x ^ 2}, {x, y}∈Circle[], Method -> "TrapezoidalRule", WorkingPrecision -> 15] Out[1]= 1.54109696443417 In[2]:= NLineIntegrate[{-Sin[y] ^ 3, x ^ 2}, {x, y}∈Circle[], Method -> "NewtonCotesRule", WorkingPrecision -> 15] Out[2]= 1.54109696015561 In[3]:= NLineIntegrate[{-Sin[y] ^ 3, x ^ 2}, {x, y}∈Circle[], Method -> "ClenshawCurtisRule", WorkingPrecision -> 15] Out[3]= 1.54109696438290 ``` With the default option: ```wl In[4]:= NLineIntegrate[{-Sin[y] ^ 3, x ^ 2}, {x, y}∈Circle[], WorkingPrecision -> 15] Out[4]= 1.54109696445681 ``` Compare to the truncated exact result: ```wl In[5]:= LineIntegrate[{-Sin[y] ^ 3, x ^ 2}, {x, y}∈Circle[]] Out[5]= (1/2) π (3 BesselJ[1, 1] - BesselJ[1, 3]) In[6]:= %//N[#, 15]& Out[6]= 1.54109696445681 ``` #### MinRecursion (1) The option ``MinRecursion`` forces a minimum number of subdivisions: ```wl In[1]:= NLineIntegrate[Exp[-100(x ^ 2 + y ^ 2)], {x, y}∈Line[{{-60, -60}, {60, 60}}]] ``` NIntegrate::ncvb: NIntegrate failed to converge to prescribed accuracy after 9 recursive bisections in \[FormalT] near {\[FormalT]} = {0.499984}. NIntegrate obtained 0.08862269273656814\` and 0.00006253563251259912\` for the integral and error estimates. ```wl Out[1]= 0.0886227 In[2]:= NLineIntegrate[Exp[-100(x ^ 2 + y ^ 2)], {x, y}∈Line[{{-60, -60}, {60, 60}}], MinRecursion -> 5] Out[2]= 0.177245 ``` Compare to the exact result: ```wl In[3]:= LineIntegrate[Exp[-100(x ^ 2 + y ^ 2)], {x, y}∈Line[{{-60, -60}, {60, 60}}]] Out[3]= (1/10) Sqrt[π] Erf[600 Sqrt[2]] In[4]:= N[%] Out[4]= 0.177245 ``` #### PrecisionGoal (1) The option ``PrecisionGoal`` sets the relative tolerance in the integration: ```wl In[1]:= exact = LineIntegrate[{Exp[-y ^ 2], Sin[x]}, {x, y}∈Circle[{0, 0}, {1, 2}]] Out[1]= 4 π BesselJ[1, 1] In[2]:= NLineIntegrate[{Exp[-y ^ 2], Sin[x]}, {x, y}∈Circle[{0, 0}, {1, 2}], PrecisionGoal -> 20, MaxRecursion -> 20, WorkingPrecision -> 20] - exact Out[2]= 0``19.257287532566455 ``` With default settings: ```wl In[3]:= NLineIntegrate[{Exp[-y ^ 2], Sin[x]}, {x, y}∈Circle[{0, 0}, {1, 2}]] - exact Out[3]= 4.440892098500626`*^-15 ``` #### WorkingPrecision (2) If a ``WorkingPrecision`` is specified, the computation is done at that working precision: ```wl In[1]:= NLineIntegrate[1, {x, y}∈Circle[]] Out[1]= 6.28319 In[2]:= NLineIntegrate[1, {x, y}∈Circle[], WorkingPrecision -> 16] Out[2]= 6.283185307179586 ``` --- The result has finite precision if the integrand has a finite precision: ```wl In[1]:= NLineIntegrate[{-y, 0.567x}, {x, y}∈Line[{{0, 0}, {2, 4}}]] Out[1]= -1.732 ``` ### Applications (27) #### College Calculus (10) Line integral of a function $f$ over a line segment: ```wl In[1]:= f = Exp[x] y ^ 2 z ^ (3 / 2); In[2]:= reg = Line[{{-1, 3, 1}, {1, 7, 3}}]; In[3]:= NLineIntegrate[f, {x, y, z}∈reg] Out[3]= 750.123 ``` --- Line integral of a vector field over a curve: ```wl In[1]:= f = {x Sin[y], y Cos[z], z Sin[x]}; In[2]:= reg = ParametricRegion[{Cos[t], Sin[t] ^ 2, -Cos[t]}, {{t, 0, Pi}}]; In[3]:= NLineIntegrate[f, {x, y, z}∈reg] Out[3]= -0.602337 ``` --- Mass of a thin circular wire of radius 1 with linear density $\rho$ : ```wl In[1]:= ρ = BesselJ[1, x ^ 2 / 2] * y ^ 2; In[2]:= reg = Circle[{0, 0}, 1]; In[3]:= NLineIntegrate[ρ, {x, y}∈reg] Out[3]= 0.194443 ``` --- Work done by the force field $f$ on a particle that moves along a line segment: ```wl In[1]:= f = {x - y ^ 3, y - z ^ 3, z - x ^ 3}; In[2]:= reg = Line[{{0, 0, 0}, {1, 2, 3}}]; In[3]:= NLineIntegrate[f, {x, y, z}∈reg] Out[3]= -9.25 ``` --- Line integral of a vector field along a path: ```wl In[1]:= f = {Cos[x], Sin[y]}; In[2]:= reg = Line[{{0, 0}, {3, 0}, {3, 3}, {0, 3}, {0, 0}}]; In[3]:= NLineIntegrate[f, {x, y}∈reg] Out[3]= 4.440892098500626`*^-16 ``` --- Line integral of a vector field along a curve: ```wl In[1]:= f = {ArcTan[x ^ 2], y ^ 2}; In[2]:= reg = ParametricRegion[{t, t ^ 2}, {{t, 0, 1}}]; In[3]:= NLineIntegrate[f, {x, y}∈reg] Out[3]= 0.631236 ``` --- Work done by the force $f$ as a particle moves along the curve $\text{reg}$ : ```wl In[1]:= f = {x ^ 2, y}; In[2]:= reg = ParametricRegion[{t - Sin[t], 1 - Cos[t]}, {{t, 0, 2Pi}}]; In[3]:= NLineIntegrate[f, {x, y}∈reg] Out[3]= 82.6834 ``` --- Line integral of a vector field along the unit circle centered at the origin: ```wl In[1]:= f = {x + y, x ^ 2}; In[2]:= NLineIntegrate[f, {x, y}∈Circle[]] Out[2]= -3.14159 ``` --- Line integral of a vector field along a circle of radius 2 centered at the origin: ```wl In[1]:= f = {x ^ 2 / (x ^ 2 + y ^ 2), (y ^ 2 - x ^ 2) / (x ^ 2 + y ^ 2)}; In[2]:= NLineIntegrate[f, {x, y}∈Circle[{0, 0}, 2]]//Quiet Out[2]= 1.4988010832439613`*^-15 ``` --- Numerical value of the line integral of a vector field over a path: ```wl In[1]:= f = {Exp[Sin[x ^ 2]], Exp[Sqrt[x ^ 2 + y ^ 2]]}; In[2]:= reg = ParametricRegion[{Sin[t ^ 2], Cos[t] ^ 3}, {{t, 0, 2Pi}}]; In[3]:= NLineIntegrate[f, {x, y}∈reg, WorkingPrecision -> 16] Out[3]= 1.602748156679319 ``` #### Lengths (3) Circumference of a circle: ```wl In[1]:= NLineIntegrate[1, {x, y}∈Circle[]] Out[1]= 6.28319 ``` --- Perimeter of a cardioid using a line integral: ```wl In[1]:= reg = ParametricRegion[{(2Cos[t] - Cos[2t]), (2Sin[t] - Sin[2t])}, {{t, 0, 2Pi}}]; In[2]:= RegionPlot[reg, ImageSize -> Small] Out[2]= [image] In[3]:= NLineIntegrate[1, {x, y}∈reg] Out[3]= 16. ``` The length can also be calculated with ``RegionMeasure`` : ```wl In[4]:= RegionMeasure[reg] Out[4]= 16 ``` --- Perimeter of an astroid: ```wl In[1]:= reg = ParametricRegion[{Cos[t] ^ 3, Sin[t] ^ 3}, {{t, 0, 2Pi}}]; In[2]:= RegionPlot[reg, ImageSize -> Small] Out[2]= [image] In[3]:= NLineIntegrate[1, {x, y}∈reg] Out[3]= 6. ``` #### Areas (5) Area of an ellipse with semiaxes of length 2 and 3, calculated using a line integral: ```wl In[1]:= NLineIntegrate[{-y / 2, x / 2}, {x, y}∈Circle[{0, 0}, {2, 3}]] Out[1]= 18.8496 ``` --- Area of the right-hand loop of the lemniscate $r^2=\cos (2 \phi )$ computed using a line integral: ```wl In[1]:= f = {-y / 2, x / 2}; In[2]:= reg = ParametricRegion[{Sqrt[Cos[2ϕ]]Cos[ϕ], Sqrt[Cos[2ϕ]]Sin[ϕ]}, {{ϕ, -Pi / 4, Pi / 4}}]; In[3]:= Show[VectorPlot[f, {x, 0, 1}, {y, -1 / 2, 1 / 2}, ...], ...] Out[3]= [image] In[4]:= NLineIntegrate[f, {x, y}∈reg] Out[4]= 0.5 ``` --- Area of the epicycloid of parameters $R=2$ and $r=1/2$ : ```wl In[1]:= R = 2; In[2]:= r = 1 / 2; In[3]:= reg = ParametricRegion[{(R + r)Cos[t] - r Cos[(R + r) / r * t], (R + r)Sin[t] - r Sin[(R + r) / r * t]}, {{t, 0, 2Pi}}]; In[4]:= Show[VectorPlot[{-y / 2, x / 2}, {x, -3, 3}, {y, -3, 3}, ...], ...] Out[4]= [image] In[5]:= NLineIntegrate[{-y / 2, x / 2}, {x, y}∈reg] Out[5]= 23.5619 ``` --- Area of the cardioid using a line integral: ```wl In[1]:= reg = ParametricRegion[{(2Cos[t] - Cos[2t]), (2Sin[t] - Sin[2t])}, {{t, 0, 2Pi}}]; In[2]:= RegionPlot[reg, ImageSize -> Small] Out[2]= [image] In[3]:= NLineIntegrate[{-y / 2, x / 2}, {x, y}∈reg] Out[3]= 18.8496 ``` --- Area of an astroid using a line integral: ```wl In[1]:= reg = ParametricRegion[{Cos[t] ^ 3, Sin[t] ^ 3}, {{t, 0, 2Pi}}]; In[2]:= RegionPlot[reg, ImageSize -> Small] Out[2]= [image] In[3]:= NLineIntegrate[{-y / 2, x / 2}, {x, y}∈reg] Out[3]= 1.1781 ``` #### Work by a Force (4) Work done by a force $f$ force as an object is moved on a straight line: ```wl In[1]:= f = {0, 0, -1}; In[2]:= NLineIntegrate[f, {x, y, z}∈Line[{{0, 0, 0}, {1, 2, 3}}]] Out[2]= -3. ``` --- Work done by the electric force as a charged particle of charge $q$ is moved from ``{1, 1, 0}`` to ``{2, 2, 0}`` in the electric field of a charged infinite wire of charge density $\lambda$ : ```wl In[1]:= Subscript[ϵ, 0] = 8.854 * 10 ^ -12; In[2]:= q = 10 ^ -6; In[3]:= λ = 10 ^ -5; In[4]:= f = {(λ * q * x/2Pi Subscript[ϵ, 0](x ^ 2 + y ^ 2)), (λ * q * y/2Pi Subscript[ϵ, 0](x ^ 2 + y ^ 2)), 0}; In[5]:= NLineIntegrate[f, {x, y, z}∈Line[{{1, 1, 0}, {2, 2, 0}}]] Out[5]= 0.124597 ``` --- Work done by an elastic force directed toward the origin as a quarter of an ellipse is traced: ```wl In[1]:= f = {-2x, -2y}; In[2]:= reg = Circle[{0, 0}, {1, 2}, {0, Pi / 2}]; In[3]:= NLineIntegrate[f, {x, y}∈reg] Out[3]= -3. ``` --- Work of the electric force as a charge $q$ is moved along the $x$ axis from $x_0$ to infinity in the electric field of a charge $Q$ : ```wl In[1]:= Subscript[ϵ, 0] = 8.854 * 10 ^ -12; In[2]:= Q = 2. * 10 ^ -5; In[3]:= q = 10 ^ -5; In[4]:= Subscript[x, 0] = 1; In[5]:= f = {(Q * q * x/4π Subscript[ϵ, 0](x ^ 2 + y ^ 2 + z ^ 2) ^ (3 / 2)), (Q * q * y/4π Subscript[ϵ, 0](x ^ 2 + y ^ 2 + z ^ 2) ^ (3 / 2)), (Q * q * z/4π Subscript[ϵ, 0](x ^ 2 + y ^ 2 + z ^ 2) ^ (3 / 2))}; In[6]:= reg = ParametricRegion[{Subscript[x, 0] + t, 0, 0}, {{t, 0, Infinity}}]; In[7]:= NLineIntegrate[f, {x, y, z}∈reg] Out[7]= 1.79755 ``` #### Centroids (2) Mass of a closed semicircular wire of radius 2 and unit linear density: ```wl In[1]:= reg = Disk[{0, 0}, 2, {0, Pi}]; In[2]:= m = NLineIntegrate[1, {x, y}∈reg] Out[2]= 10.2832 ``` $x$ coordinate of the center of mass: ```wl In[3]:= (1 / m) * NLineIntegrate[x, {x, y}∈reg, AccuracyGoal -> 5] Out[3]= 7.631588740349012`*^-17 ``` $y$ coordinate of the center of mass: ```wl In[4]:= (1 / m) * NLineIntegrate[y, {x, y}∈reg, AccuracyGoal -> 5] Out[4]= 0.777969 ``` --- Moments of inertia of a helix-shaped wire of unit linear density: ```wl In[1]:= reg = ParametricRegion[{Cos[t], Sin[t], t / 10}, {{t, 0, 6Pi}}]; In[2]:= ParametricPlot3D[{Cos[t], Sin[t], t / 10}, {t, 0, 6Pi}, ImageSize -> Small] Out[2]= [image] ``` About the $x$ axis: ```wl In[3]:= NLineIntegrate[(y ^ 2 + z ^ 2), {x, y, z}∈reg] Out[3]= 31.9076 ``` About the $y$ axis: ```wl In[4]:= NLineIntegrate[(x ^ 2 + z ^ 2), {x, y, z}∈reg] Out[4]= 31.9076 ``` About the $z$ axis: ```wl In[5]:= NLineIntegrate[(x ^ 2 + y ^ 2), {x, y, z}∈reg] Out[5]= 18.9436 ``` #### Classical Theorems (3) A vector field is *conservative* if its line integral depends only on the values at the endpoints, not on the path: ```wl In[1]:= f = {3 x^2 y, x^3}; ``` The field $f$ is the gradient of a scalar function $g$ : ```wl In[2]:= g = x ^ 3 y; In[3]:= f == Grad[g, {x, y}] Out[3]= True ``` All gradients of scalar fields are conservative. For example, the line integral of $f$ over the curve is: ```wl In[4]:= reg = ParametricRegion[{t ^ 2, Exp[t]}, {{t, 0, 1}}]; In[5]:= NLineIntegrate[f, {x, y}∈reg] Out[5]= 2.71828 ``` This is the same as the difference of the values of $g$ at the endpoints of the curve: ```wl In[6]:= (g /. {x -> t ^ 2, y -> Exp[t]} /. t -> 1) - (g /. {x -> t ^ 2, y -> Exp[t]} /. t -> 0)//N Out[6]= 2.71828 ``` --- Green's theorem. The line integral of the vector field $f$ over a closed curve is: ```wl In[1]:= f = {x ^ 4, x * y ^ 2 + 1}; In[2]:= NLineIntegrate[f, {x, y}∈Line[{{0, 0}, {1, 0}, {0, 1}, {0, 0}}]] Out[2]= 0.0833333 ``` This can be related to a surface integral of $g$ over the region enclosed by the curve, where $g$ is defined as: ```wl In[3]:= g = D[f[[2]], x] - D[f[[1]], y]; In[4]:= NIntegrate[g, {x, y}∈Triangle[{{0, 0}, {1, 0}, {0, 1}}]] Out[4]= 0.0833333 ``` --- Stokes's theorem. The line integral of a vector field $f$ along a closed line in three dimensions is: ```wl In[1]:= f = {-y + x y ^ 2, x, z}; In[2]:= reg = ParametricRegion[{Cos[t], Sin[t], 0}, {{t, 0, 2Pi}}]; In[3]:= NLineIntegrate[f, {x, y, z}∈reg] Out[3]= 6.28319 ``` This is equal to the surface integral of the ``Curl`` of $f$ on any surface having the curve as its boundary: ```wl In[4]:= reg2 = ParametricRegion[{Cos[t]Cos[p], Cos[t]Sin[p], Sin[t]}, {{p, 0, 2Pi}, {t, 0, Pi / 2}}]; In[5]:= NSurfaceIntegrate[Curl[f, {x, y, z}], {x, y, z}∈reg2] Out[5]= 6.28319 ``` The surface integral across a different surface with the same boundary is the same: ```wl In[6]:= NSurfaceIntegrate[Curl[f, {x, y, z}], {x, y, z}∈ParametricRegion[{r Cos[t], r Sin[t], 0}, {{r, 0, 1}, {t, 0, 2Pi}}]] Out[6]= 6.28319 ``` ### Properties & Relations (5) Apply ``N[LineIntegrate[…]]`` to obtain a numerical solution if the symbolic calculation fails: ```wl In[1]:= f = Sin[x + y + z]; In[2]:= reg = ParametricRegion[{t ^ 2, t ^ 3, t ^ 4}, {{t, 0, 1}}]; In[3]:= LineIntegrate[f, {x, y, z}∈reg] Out[3]= LineIntegrate[Sin[x + y + z], {x, y, z}∈ParametricRegion[{{t^2, t^3, t^4}, 0 ≤ t ≤ 1}, {t}]] In[4]:= N[LineIntegrate[f, {x, y, z}∈reg]] Out[4]= 1.16617 In[5]:= NLineIntegrate[f, {x, y, z}∈reg] Out[5]= 1.16617 ``` --- Find the center of mass of a triangular wire of unit linear density: ```wl In[1]:= reg = Line[{{0, 0}, {1, 0}, {1, 1}, {0, 0}}]; In[2]:= RegionPlot[reg, ImageSize -> Small] Out[2]= [image] ``` Find the total mass: ```wl In[3]:= m = NLineIntegrate[1, {x, y}∈reg] Out[3]= 3.41421 ``` Find the $x$ component of the center of mass: ```wl In[4]:= 1 / m * NLineIntegrate[x, {x, y}∈reg] Out[4]= 0.646447 ``` Find the $y$ component: ```wl In[5]:= 1 / m * NLineIntegrate[y, {x, y}∈reg]//Quiet Out[5]= 0.353553 ``` The center of mass can also be obtained using ``RegionCentroid`` : ```wl In[6]:= RegionCentroid[reg]//N Out[6]= {0.646447, 0.353553} ``` --- Find the moment of inertia around the $z$ axis of a circular wire of unit linear density centered at the origin in the $x$ - $y$ plane: ```wl In[1]:= reg = ParametricRegion[{Cos[t], Sin[t], 0}, {{t, 0, 2Pi}}]; In[2]:= NLineIntegrate[(x ^ 2 + y ^ 2), {x, y, z}∈reg] Out[2]= 6.28319 ``` The answer can also be computed with ``MomentOfInertia`` : ```wl In[3]:= MomentOfInertia[reg, {0, 0, 0}, {0, 0, 1}]//N Out[3]= 6.28319 ``` --- Find the length of an epicycloid: ```wl In[1]:= reg = ParametricRegion[{6 Cos[t] - Cos[6 t], 6 Sin[t] - Sin[6 t]}, {{t, 0, 2Pi}}]; In[2]:= RegionPlot[reg, ImageSize -> Small] Out[2]= [image] In[3]:= NLineIntegrate[1, {x, y}∈reg]//Quiet Out[3]= 48. ``` The same answer can be obtained using ``ArcLength`` : ```wl In[4]:= ArcLength[{6 Cos[t] - Cos[6 t], 6 Sin[t] - Sin[6 t]}, {t, 0, 2Pi}] Out[4]= 48 ``` --- Find the area of an ellipse: ```wl In[1]:= reg = Circle[{0, 0}, {1 / Sqrt[2], 1 / Sqrt[3]}]; In[2]:= NLineIntegrate[{-y / 2, x / 2}, {x, y}∈reg] Out[2]= 1.28255 ``` The result can be obtained using ``RegionMeasure`` : ```wl In[3]:= RegionMeasure[Disk[{0, 0}, {1 / Sqrt[2], 1 / Sqrt[3]}]] Out[3]= (π/Sqrt[6]) ``` ### Neat Examples (2) Length of a catenary: ```wl In[1]:= reg = ParametricRegion[{t, Cosh[t]}, {{t, -1, 1}}]; In[2]:= ParametricPlot[{t, Cosh[t]}, {t, -1, 1}, ImageSize -> Small] Out[2]= [image] In[3]:= NLineIntegrate[1, {x, y}∈reg] Out[3]= 2.3504 ``` --- Integral of a vector field over a Clelia curve: ```wl In[1]:= f = {x ^ 2 + y, y ^ 2 + z, z ^ 2 + x}; In[2]:= reg = ParametricRegion[{Cos[t]Cos[3t], Cos[t]Sin[3t], Sin[t]}, {{t, 0, 2Pi}}]; In[3]:= Show[VectorPlot3D[f, {x, -1, 1}, {y, -1, 1}, {z, -1, 1}, Rule[...]], ...] Out[3]= [image] In[4]:= NLineIntegrate[f, {x, y, z}∈reg] Out[4]= -4.71239 ``` ## See Also * [`LineIntegrate`](https://reference.wolfram.com/language/ref/LineIntegrate.en.md) * [`SurfaceIntegrate`](https://reference.wolfram.com/language/ref/SurfaceIntegrate.en.md) * [`NSurfaceIntegrate`](https://reference.wolfram.com/language/ref/NSurfaceIntegrate.en.md) * [`ContourIntegrate`](https://reference.wolfram.com/language/ref/ContourIntegrate.en.md) * [`NContourIntegrate`](https://reference.wolfram.com/language/ref/NContourIntegrate.en.md) * [`Integrate`](https://reference.wolfram.com/language/ref/Integrate.en.md) * [`NIntegrate`](https://reference.wolfram.com/language/ref/NIntegrate.en.md) * [`Region`](https://reference.wolfram.com/language/ref/Region.en.md) * [`ParametricRegion`](https://reference.wolfram.com/language/ref/ParametricRegion.en.md) * [`ArcLength`](https://reference.wolfram.com/language/ref/ArcLength.en.md) * [`Grad`](https://reference.wolfram.com/language/ref/Grad.en.md) ## Related Guides * [Vector Analysis](https://reference.wolfram.com/language/guide/VectorAnalysis.en.md) * [Symbolic Vectors, Matrices and Arrays](https://reference.wolfram.com/language/guide/SymbolicArrays.en.md) ## History * [Introduced in 2024 (14.0)](https://reference.wolfram.com/language/guide/SummaryOfNewFeaturesIn140.en.md) \| [Updated in 2025 (14.2)](https://reference.wolfram.com/language/guide/SummaryOfNewFeaturesIn142.en.md)