Volumes and double integrals

Purpose

The purpose of this lab is to acquaint you with using Maple to do double integrals and to help you visualize a certain class of solids, whose volumes can be computed via double integrals.

Background

Computing double integrals in Maple

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

or

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.

  > int(int(x^2+y^2,x=-2..1),y=0..1);

  > int(int(x^2+y^2,y=0..1),x=-2..1);

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

  > int(int(x^2+y^2,y=-sqrt(1-(x-1)^2)..sqrt(1-(x-1)^2)),x=0..2);

Visualizing with Maple

Maple can also help you visualize the solids of which you are computing volumes. Here we go through several Maple commands related to the solid of the last example. In the sample Maple session shown below, the first step, after loading the plots package, is to plot the region in the x-y plane. Note that Maple can 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,

  > plot3d(h(x,y),x=a..b,y=f(x)..g(x);

assuming that forms the lower boundary of the region. Read the help for plot3d for more details. Note the 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 .

The next two commands in the Maple session show cross-sections of the solid for x=1 and , and the last set of three commands generates a plot of the lateral boundary of the solid and plots of the top and bottom of the solid, and displays the three plots with the display3d command.

  > with(plots):

  > plot3d(0,x=0..2,y=-sqrt(1-(x-1)^2)..sqrt(1-(x-1)^2));

  > f := (x,y) -> x^2+y^2 ;

  > plot3d([1,t,s*f(1,t)],t=-1..1,s=0..1,labels=[x,y,z]);

  > p1 := plot3d([1+cos(t),sin(t),s*f(1+cos(t),sin(t))],t=0..2*Pi,s=0..1):

  > p2 := plot3d(0,f(x,y),x=0..2,y=-sqrt(1-(x-1)^2)..sqrt(1-(x-1)^2)):

  > display3d(p1,p2);

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. Note especially that the commands defining p1 and p2 end in colons, (:). This is so the output of the plot3d commands doesn't clutter up your worksheet.

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 , 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 .

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.

  > plot3d([x(t),y(t),s*f(x(t),y(t))],t=a..b,s=0..1);

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
  > plot3d(f(x,y),x=0..2,y=-sqrt(1-(x-1)^2)..sqrt(1-(x-1)^2));

Exercises

1. Use Maple to compute the following double integrals.

   1. 

   2. 

   3. 

2. Explain how the Maple commands used for the first example in the Background section follow from the theory at the end of that section.
3. For the following functions and regions, compute the volume between the region and the graph of the function and produce a plot of the solid so generated.
   1. Between the circle and the function .
   2. Between the region in the x-y plane containing the origin and bounded by the two curves and and the function .
4. Find the volume of the first octant part of the solid bounded by the cylinders and . Also, generate a three-dimensional plot of the surface of the solid.