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Student[MultivariateCalculus]

 MultiInt
 return the integral of a function defined over a region

 Calling Sequence MultiInt(f(x,y), x=a..b, y=c..d, opts) MultiInt(f(x,y,z), x=a..b, y=c..d, z=e..f, opts) MultiInt(f(vars), [vars]=region, opts)

Parameters

 f(x,y), f(x,y,z), f(vars) - algebraic expressions x, y, z - name; specify the independent variables a, b, c, d, e, f - algebraic; limits of integration vars - sequence of names; specify the independent variables region - one of the possible regions of integration as described below opts - (optional) equation(s) of the form option=value where option is coordinates or output; specify output options

Description

 • The MultiInt command returns the integral of a function over a region in space.
 • The opts argument can contain any of the following equations.
 coordinates = cartesian[x,y], polar[r,theta] (2-D), cartesian[x,y,z], cylindrical[r,theta,z], or spherical[r,phi,theta]
 Specifies the coordinate system being used. The first variable of polar, cylindrical, and spherical is assumed to be the radial component. For spherical coordinates [r, phi,theta], the second coordinate is the angle measured down from the vertical.
 The default is cartesian.
 Note: The coordinates option is not valid for the third calling sequence.
 output = value, integral, or steps
 Specifies the return value of the function.
 output = value specifies that the value of the integral is returned. This is the default.
 output = integral specifies that the inert form of the integral is shown.
 output = steps specifies that the intermediate steps of solving the multiple integral are shown.
 • The third calling sequence specifies a region of integration using the parameter [vars]=region. The possible regions of integration are Circle, Ellipse, Parallelepiped, Rectangle, Region, Sector, Sphere, Tetrahedron, and Triangle.
 – Circle(center, radius) The first parameter, center, is the center of the circle and must be a two-dimensional Vector with algebraic entries. The second parameter, radius, is the radius of the circle and must be an algebraic expression.
 – Ellipse(center, major, minor, angle) The first parameter, center, is the center of the ellipse and must be a two-dimensional Vector with algebraic entries. 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 algebraic expression. 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 vars.
 – Parallelepiped(r1, ..., rk) This specifies a k-dimensional region of integration. Each ri is a range of the form lowerlimit..upperlimit and the expressions in these ranges must be algebraic and cannot depend on the variables of integration. The range ri specifies the integration interval for the ith variable of integration.
 – Rectangle(r1, r2) This is a specialization of Parallelepiped to ${ℝ}^{2}$ . In the x-y-plane, the integration has the order dx dy, and range r1 can be of the form x1(y)..x2(y), in contrast to the restrictions on ranges in Parallelepiped.
 – Region(r1, ..., rk) This specifies a k-dimensional region of integration. Each ri is a range of the form lowerlimit..upperlimit and the expressions in these ranges must be algebraic and the ith range can only depend on the first i-1 variables. The integral over the region is transformed into an iterated integral. The range ri specifies the integration interval for the ith variable of integration.
 – Sector(object, start, finish) The first parameter of Sector, object, must be a Circle or Ellipse structure. The parameters start and finish must be algebraic expressions. This represents the sector 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 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 $\frac{\mathrm{\pi }}{4}$ (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 region of the ellipse inside 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.
 – Sphere(center, radius) The first parameter, center, is the center of the sphere and must be a three-dimensional Vector with algebraic entries. The second parameter, radius, is the radius of the sphere and must be an algebraic expression.
 – Tetrahedron(v1, v2, v3, v4) Each vi is a three-dimensional Vector (with algebraic entries). These represent the vertices of the tetrahedron. If coordinate system attributes are specified on these Vectors, each vertex is interpreted in its own coordinate system.
 – Triangle(v1, v2, v3) Each vi is a two-dimensional Vector (with algebraic entries). These represent the vertices of the triangle. If coordinate system attributes are specified on these Vectors, each vertex is interpreted in its own coordinate system.

Examples

 > $\mathrm{with}\left(\mathrm{Student}[\mathrm{MultivariateCalculus}]\right):$
 > $\mathrm{MultiInt}\left(3{x}^{2}+3{y}^{2},x=1..4,y=-1..6\right)$
 ${1092}$ (1)
 > $\mathrm{MultiInt}\left(r,r=1..4,\mathrm{θ}=0..\frac{\mathrm{π}}{2},\mathrm{coordinates}={\mathrm{polar}}_{r,\mathrm{θ}}\right)$
 $\frac{{21}}{{2}}{}{\mathrm{π}}$ (2)
 > $\mathrm{MultiInt}\left(r+\mathrm{θ}+z,r=0..2,\mathrm{θ}=2..4,z=\frac{\mathrm{π}}{2}..\mathrm{π},\mathrm{coordinates}={\mathrm{cylindrical}}_{r,\mathrm{θ},z}\right)$
 $\frac{{26}}{{3}}{}{\mathrm{π}}{+}\frac{{3}}{{2}}{}{{\mathrm{π}}}^{{2}}$ (3)
 > $\mathrm{MultiInt}\left(r,r=0..\mathrm{π},\mathrm{θ}=0..\mathrm{π},\mathrm{φ}=0..\mathrm{π},\mathrm{coordinates}={\mathrm{spherical}}_{r,\mathrm{θ},\mathrm{φ}}\right)$
 $\frac{{1}}{{2}}{}{{\mathrm{π}}}^{{5}}$ (4)
 > $\mathrm{MultiInt}\left({x}^{2}+{y}^{2}+z,z=-2..4+{y}^{2},y=x-1..x+6,x=2..4,\mathrm{output}=\mathrm{integral}\right)$
 ${{∫}}_{{2}}^{{4}}{{∫}}_{{x}{-}{1}}^{{x}{+}{6}}{{∫}}_{{-}{2}}^{{{y}}^{{2}}{+}{4}}\left({{x}}^{{2}}{+}{{y}}^{{2}}{+}{z}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{z}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{y}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{x}$ (5)
 > $\mathrm{MultiInt}\left({x}^{2}+{y}^{2},\left[x,y\right]=\mathrm{Circle}\left(⟨0,0⟩,r\right)\right)$
 $\frac{{1}}{{2}}{}{\mathrm{π}}{}{{r}}^{{4}}$ (6)
 > $\mathrm{MultiInt}\left(xy,\left[x,y\right]=\mathrm{Triangle}\left(⟨0,0⟩,⟨1,0⟩,⟨0,1⟩\right)\right)$
 $\frac{{1}}{{24}}$ (7)
 > $\mathrm{MultiInt}\left({ⅇ}^{-{x}^{2}-{y}^{2}-{z}^{2}},\left[x,y,z\right]=\mathrm{Sphere}\left(⟨0,0,0⟩,r\right)\right)$
 ${\mathrm{π}}{}\left(\sqrt{{\mathrm{π}}}{}{\mathrm{erf}}{}\left({r}\right){}{{ⅇ}}^{{{r}}^{{2}}}{-}{2}{}{r}\right){}{{ⅇ}}^{{-}{{r}}^{{2}}}$ (8)
 > $\mathrm{MultiInt}\left(\mathrm{sin}\left(x\right)\mathrm{cos}\left(y\right)\mathrm{tan}\left(z\right),\left[x,y,z\right]=\mathrm{Parallelepiped}\left(0..\mathrm{π},0..\frac{\mathrm{π}}{2},0..\frac{\mathrm{π}}{4}\right)\right)$
 ${\mathrm{ln}}{}\left({2}\right)$ (9)
 > $\mathrm{MultiInt}\left(1,\left[x,y\right]=\mathrm{Region}\left(0..1,{x}^{2}..x\right)\right)$
 $\frac{{1}}{{6}}$ (10)
 > $\mathrm{MultiInt}\left(x+y+z,\left[x,y,z\right]=\mathrm{Tetrahedron}\left(⟨0,0,0⟩,⟨1,0,0⟩,⟨0,1,0⟩,⟨0,0,1⟩\right),\mathrm{output}=\mathrm{integral}\right)$
 ${{∫}}_{{0}}^{{1}}{{∫}}_{{0}}^{{1}{-}{t}}{{∫}}_{{0}}^{\frac{\left({-}{1}{+}{t}\right){}\left({x}{-}{1}{+}{t}\right)}{{1}{-}{t}}}\left({x}{+}{y}{+}{t}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{y}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{x}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{t}$ (11)
 > $\mathrm{MultiInt}\left(1,\left[x,y\right]=\mathrm{Ellipse}\left(\frac{{x}^{2}}{4}+\frac{{y}^{2}}{9}-1\right)\right)$
 ${6}{}{\mathrm{π}}$ (12)

You can specify an equivalent integration by giving the parameters of the ellipse directly.

 > $\mathrm{MultiInt}\left(1,\left[x,y\right]=\mathrm{Ellipse}\left(⟨0,0⟩,3,2,0\right)\right)$
 ${6}{}{\mathrm{π}}$ (13)
 > $\mathrm{MultiInt}\left({y}^{2},\left[x,y\right]=\mathrm{Sector}\left(\mathrm{Circle}\left(⟨0,0⟩,r\right),0,\mathrm{π}\right)\right)$
 $\frac{{1}}{{8}}{}{\mathrm{π}}{}{{r}}^{{4}}$ (14)
 > $\mathrm{MultiInt}\left(x,\left[x,y\right]=\mathrm{Sector}\left(\mathrm{Ellipse}\left(\frac{{x}^{2}}{4}+\frac{{y}^{2}}{9}-1\right),0,\frac{\mathrm{π}}{2}\right)\right)$
 ${4}$ (15)

Compatibility

 • The region parameter was introduced in Maple 2016.