The ApproximateInt(f(x,y),x=a..b,y=c..d) calling sequence returns an approximation to the integral of $f\left(x\,y\right)$.

The ApproximateIntTutor routine offers equivalent capabilities to ApproximateInt in a tutor interface. See the Student[MultivariateCalculus][ApproximateIntTutor] help page.

The opts argument can contain any of the following equations that set output options.

coordinates = cartesian or polar

Determines whether the integral is calculated in cartesian or polar coordinates. In polar coordinates, the first variable is assumed to be the radial component. The default is cartesian.

frames = posint..posint

For example, frames = a..b.

Specifies the number of frames in the animation, b-a+1. Each frame of the animation partitions the plot according to the number in the range a..b corresponding to the current frame. The first frame partitions the domain into an a x a grid. The last frame partitions the domain into a b x b grid. This option is ignored unless output = animation. The default is $3..7$.

functionoptions = list

Specifies the plot options for plotting the function $f\left(x\,y\right)$. For more information on plotting options, see plot3d/options.

method = midpoint, upper, lower, or random

Calculates the approximation of the integral using the midpoint, upper, lower, or random point Riemann sum. The default is midpoint.

output = value, plot, or animation

This option controls the return value of the function.

* output = value specifies that the value of the approximation is returned. Plot options are ignored if output = value. The default is output=value.

* output = plot specifies that a plot displays, which shows the expression and an approximation to the integral.

* output = animation specifies that an animation displays, which shows the expression and approximations using a sequence of partitions each of which is a refinement of its predecessor. Note: To play an animation in this help page, right-click (Control-click, on Macintosh) the plot to display the context menu.  Select Animation > Play.

partition = [posint, posint]

The partition option controls the partition of the interval. By default, the interval is partitioned by $\left[5\,5\right]$.

prismoptions = list

Specifies the plot options for plotting the approximation to the integral of $f\left(x\,y\right)$. For more information on plotting options, see plot3d/options.

showfunction = true or false

Determines whether the function $f\left(x\,y\right)$ is plotted. The default is true.

caption = anything

A caption for the plot.

The default caption is constructed from the parameters and the command options. caption = "" disables the default caption. For more information about specifying a caption, see plot/typesetting.

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

$\mathrm{ApproximateInt}\left(x+y^2\,x=1..4\,y=-1..6\right)$

$266.0700000$

$\mathrm{ApproximateInt}\left(x^2+y^2\,x=-4..4\,y=-6..6\,\mathrm{output}=\mathrm{animation}\,\mathrm{partition}=\left[6\,6\right]\,\mathrm{frames}=2..7\right)$

$\mathrm{ApproximateInt}\left(\sin \left(x\right)\sin \left(y\right)\,x=-3..3\,y=-3..3\,z=-2..2\,\mathrm{output}=\mathrm{plot}\,\mathrm{partition}=\left[5\,5\right]\right)$

$\mathrm{ApproximateInt}\left(r\,r=0..5\,t=0..\frac{\mathrm{\pi }}{2}\,\mathrm{coordinates}=\mathrm{polar}\,\mathrm{output}=\mathrm{plot}\,\mathrm{showfunction}=\mathrm{false}\,\mathrm{prismoptions}=\left[\mathrm{color}=\mathrm{green}\right]\,\mathrm{orientation}=\left[165\,80\,0\right]\right)$