Suppose that D is a rectangular region in the the x-y plane, and that is a continuous, non-negative function on D. Then the volume of the solid above D and below is given by the double integral
We learned in class that such integrals can be evaluated by either of the iterated integrals
where the rectangle D is defined by the inequalities and .
Using Maple to evaluate double integrals proceeds in exactly the same fashion. For example, the following Maple session shows the computation of
for both orders of integration.
More complicated integrals are handled similarly. Suppose that we have and we want to know the volume of the solid between the region in the x-y plane bounded by the circle and the graph of . Then an appropriate integral is
Using Maple, we calculate this as
Maple can also help you visualize the solids whose volumes you are computing. Here we go through several Maple commands related to the solid of the last example. We will plot cross-sections through the solid as well as its sides, top, and bottom. To do this in Maple, some of the plots have to be in parametric form, which means we represent a surface by three coordinate functions , , and . We also use a special trick to plot a function over a region in the plane defined by an x-interval, and two functions and with a command of the following form,
assuming that forms the lower boundary and the upper boundary of the region. Read the help for plot3d for more details. Note that the x range, x=a..b, must come first. Note also that we can plot regions in the x-y plane with a similar command simply by using .
In the Maple commands shown below, first the plots package is loaded, and then a plot of the region in the x-y plane that forms the base of the solid is created. This region is the same as the domain of integration we used above to compute the volume of the solid. Note that we give the plot a label, and end the command with a colon, not the usual semi-colon. The label is important because we want to display several plots on the same graph and the colon suppresses the output of the command. (If you want to see this output, use a semi-colon instead.) The final command defines our function.
> base := plot3d(0,x=0..2,y=-sqrt(1-(x-1)^2)..sqrt(1-(x-1)^2)):
> f := (x,y) -> x^2+y^2;
The next two Maple commands create plots of cross-sections through the solid for x=1 and . The display3d command actually displays the base of the solid and the two cross-sections.
> cross1 := plot3d([1,t,s*f(1,t)],t=-1..1,s=0..1):
> cross2 := plot3d([0.5,t*sqrt(1-1/4),s*f(0.5,t*sqrt(1-1/4))],t=-1..1,s=0..1):
The three commands below generate a plot of the lateral boundary of the solid, a plot of the top of the solid, and finally display the solid with a display3d command.
> side := plot3d([1+cos(t),sin(t),s*f(1+cos(t),sin(t))],t=0..2*Pi,s=0..1):
> top := plot3d(f(x,y),x=0..2,y=-sqrt(1-(x-1)^2)..sqrt(1-(x-1)^2)):
The display3d command may not be familiar to you, but it provides an easy way to display multiple three dimensional plots at the same time. It also allows you to set some options for the plot, as we did above to add axes and labels to the plot.
To help you understand where these Maple commands came from, first consider the cross-section for . For this value of x, we get from the equation the two roots and . So now we have the parametric description and for , of the line joining these two roots. Suppose is a point on this line. The solid contains a vertical line segment above this point until we run into . We parametrize this vertical line segment by s, so that when s=0 we are at the point and when s=1 we are at . So, finally, we represent the cross section through the solid as the parametric surface , , and .
To generate the lateral surface of the solid, we do the same sort of thing for all the points on the boundary of the region. That is, if the boundary of the region is described by a parametric curve , , then the lateral surface can be plotted with a command of the following form.
Defining and in your Maple session might be a good idea, since it will save some typing.
Plotting the top of the solid is a little easier. Suppose that the region is such that the volume can be computed with a one double integral, with x being the outer variable of integration. This means that the boundary can be decomposed into two curves and with , where ``u'' and ``l'' stand for upper and lower. For example, for our region bounded by the circle , the two curves are and , with . Then the top of our solid is generated by a Maple command of the form