The PathInt(f, domain) calling sequence computes the path integral of the function f over the path specified by domain.

If output=integral is specified, the command returns an unevaluated integral. If output=value is specified, the command returns the computed value of the integral. The default value is value.

Specify the path of integration using unevaluated function calls. The possible paths of integration are Arc, Circle, Circle3D, Ellipse, Line, LineSegments, and Path.

Arc(object, start, finish)

The first parameter of Arc, object, must be a Circle or Ellipse structure. The parameters start and finish must be of type algebraic. This represents a segment of the circumference of the specified Circle or Ellipse, starting at angle start radians and ending at angle finish radians.

For a Circle, an angle of $0$ points in the positive x direction. For example, if start is $0$ and finish is $\frac{\mathrm{\pi }}{2}$, it would represent the top-right quarter of the circumference of a circle centered anywhere.

For an Ellipse, an angle of $0$ points in the direction of one of its semimajor axes: the one in the right-half of the plane. If the Ellipse is vertical (its major axis is coincident with the y-axis), then the $0$ angle points in the negative y direction. For example, consider an ellipse centered at the origin with a rotation of Pi/4 (its major axis sits on the line y = x). If start is $-\frac{\mathrm{\pi }}{4}$ and finish is $\frac{\mathrm{\pi }}{4}$, it would represent the portion of its circumference inside the first quadrant of the plane. If the ellipse is not centered at the origin, these angles are treated as if the center of the ellipse were the origin.

Circle(center, radius)

The parameter center is the center of the circle and must have type 'Vector'(algebraic). The parameter radius is the radius of the circle and must have type algebraic.  If a coordinate system attribute is specified for center, the center is interpreted in this coordinate system.

Circle3D(cen, rad, normal)

The parameter cen is the center of the circle and must have type 'Vector'(algebraic). The parameter rad is the radius of the circle and must have type algebraic. The parameter normal is the normal to the plane in which the circle lies and must have type 'Vector'(algebraic).  If a coordinate system attribute is specified on cen, it is interpreted in that coordinate system.

Ellipse(center, major, minor, angle)

The first parameter, center, is the center of the ellipse and must have type Vector(2, algebraic). The second and third parameters, major and minor, are the lengths of the semimajor and semiminor axes of the ellipse, respectively. The last parameter angle specifies the angle through which the ellipse is rotated from its standard position, in which its major axis is coincident with the x-axis. Note: The ellipse specified by this structure is obtained by first performing the rotation on an ellipse in standard position; this rotated ellipse is then translated to the point center.

Ellipse(expr)

The parameter expr must be an expression of type algebraic. The Ellipse is specified by the equation expr = 0. The two variable names that appear in expr must be the same as those specified in the left-hand side of domain.

Line(p1, p2)

The parameters p1 and p2 must have type 'Vector'(algebraic). They represent the endpoints of the directed line segment from p1 to p2. If coordinate system attributes are specified on the points, they are interpreted in their respective coordinate systems.

LineSegments(p1, p2, ..., pk)

Similar to Line(p1, p2), the parameters pi represent the endpoints of $k-1$ line segments.  The path of integration is the collection of line segments directed from p1 to p2, p2 to p3, ..., and p(k-1) to pk. If any coordinate system attributes are specified, the points are interpreted in their respective coordinate systems.

Path(v, range, coordinate_system)

The first parameter of Path, v, is a Vector representing the components of the path. The second parameter, range, must have type {range, name=range}.  If no parameter name is specified in range, it is inferred from v.

If the optional third argument, coordinate_system, is specified, it must be an equation of the form coords=sys or coordinates=sys, where sys is a recognized coordinate system. v is interpreted in the specified coordinate system.

$\mathrm{with}\left(\mathrm{Student}\left[\mathrm{VectorCalculus}\right]\right)\:$

$\mathrm{PathInt}\left(x^2\,\left[x\,y\right]=\mathrm{Line}\left(⟨0\,0⟩\,⟨1\,1⟩\right)\right)$

$\frac{\sqrt{2}}{3}$

$\mathrm{PathInt}\left(y^2\,\left[x\,y\right]=\mathrm{LineSegments}\left(⟨0\,0⟩\,⟨1\,1⟩\,⟨1\,2⟩\right)\right)$

$\frac{\sqrt{2}}{3}+\frac{7}{3}$

$\mathrm{PathInt}\left(y\,\left[x\,y\right]=\mathrm{Path}\left(⟨\cos \left(t\right)\,\sin \left(t\right)⟩\,t=0..\mathrm{\pi }\right)\right)$

$2$

$\mathrm{PathInt}\left(y\,\left[x\,y\right]=\mathrm{Path}\left(⟨1\,t⟩\,0..\mathrm{\pi }\,'\mathrm{coords}'='\mathrm{polar}'\right)\right)$

$2$

$\mathrm{PathInt}\left(x^2+y^2\,\left[x\,y\right]=\mathrm{Circle}\left(⟨0\,0⟩\,3\right)\right)$

$54\mathrm{\pi }$

$\mathrm{PathInt}\left(x^2+y^2\,\left[x\,y\right]=\mathrm{Circle}\left(⟨0\,0⟩\,3\right)\,'\mathrm{output}'='\mathrm{integral}'\right)$

${\int }_0^{2\mathrm{\pi }}3\left(9{\cos \left(t\right)}^2+9{\sin \left(t\right)}^2\right)\sqrt{{\cos \left(t\right)}^2+{\sin \left(t\right)}^2}\ⅆt$

$\mathrm{PathInt}\left(1\,\left[x\,y\right]=\mathrm{Ellipse}\left(\frac{x^2}{4}+\frac{y^2}{9}-1\right)\right)$

$12\mathrm{EllipticE}\left(\frac{\sqrt{5}}{3}\right)$

$\mathrm{PathInt}\left(x\,\left[x\,y\right]=\mathrm{Arc}\left(\mathrm{Circle}\left(⟨0\,0⟩\,1\right)\,0\,\frac{\mathrm{\pi }}{2}\right)\right)$

$1$