In many problems on Test 3 you will need to sketch a 3D object bounded by given surfaces. You should expect to spend a good deal of time making and re-making 3D sketches when doing problems from the Preparation sheet. On the Test you will need to make those sketches without any use of software, and so it is essential that you learn to sketch in 3D “by hand”. This can only come through much practice, including many failed attempts. Work slowly; haste makes waste.

However, when preparing for the Test, you may still use Mathematica to help you decide if your sketch has been correct. So, once you have made a first reasonable draft of your sketch by hand, you may use Mathematica to confirm it. 

Below I work out a couple of examples. One of their purposes is to show you what commands you can use to sketch surfaces over regions with complex boundaries. The first example also demonstrates that even with Mathematica, you may still need to use your brain to decide if the sketch is correct.

Sec. 15.6 (8th edition), # 22
(Note:  This is not a HW problem.) 
Sketch the solid enclosed by the cylinder  x^2 + z^2 = 4 
and the planes  y= -2 and  y + z = 4 .

First, let us sketch the cylinder. We know that its axis is the  y-axis. We arbitrarily choose limits for y. The limits for x, on the other hand, appear naturally from the requirement that the expression under the square root be non-negative.

Plot3D[{(Sqrt[4-(x^2)]),-(Sqrt[4-(x^2)])},{x,-2,2},{y,-2,6},AxesLabel->{"X","Y","Z"},PlotRange->All]

Note that even if we, by mistake, take limits for  x  that are “too wide”,  that would not affect the plot:

Plot3D[{(Sqrt[4-(x^2)]),-(Sqrt[4-(x^2)])},{x,-3,3},{y,-2,6},AxesLabel->{"X","Y","Z"},PlotRange->All]

You may investigate on your own what will happen if you take limits for  x  that are “too narrow”.

Now let us include information about the two planes bounding this solid. This can be done using the option  RegionFunction  inside the  Plot3D. The syntax of  RegionFunction  requires the region to be defined by a set of inequalities. It is not difficult to guess that we should have  y >= -2. However, it is less trivial to decide if the required object satisfies  y + z >= 4  or   y + z <= 4. Therefore, we should try both:

p1=Plot3D[{(Sqrt[4-(x^2)]),-(Sqrt[4-(x^2)])},{x,-2,2},{y,-2,6},RegionFunction->Function[{x,y,z},y≥-2 && y+z≥4],AxesLabel->{"X","Y","Z"},PlotRange->All]

p2=Plot3D[{(Sqrt[4-(x^2)]),-(Sqrt[4-(x^2)])},{x,-2,2},{y,-2,6},RegionFunction->Function[{x,y,z},y≥-2 && y+z<=4],AxesLabel->{"X","Y","Z"},PlotRange->All]

To decide if the correct region is given by plot  p1  or plot  p2,  we examine the bounds for   y    in each of them. We notice that  in  p1,   y  extends on the left only to 2,  whereas in  p2,    y extends on the left to  -2. Since it is the latter bound that is prescribed in the problem, then  p2  must be the correct plot.

Sec. 15.6 (8th edition), # 28
Sketch the solid whose boundaries are set by the given limits in a triple integral:
0 <= x <= 4-y^2,  0 <= z <= 2 - y,  0 <= y <= 2.

In this case, making a plot is straightforward using command  RegionPlot3D:

RegionPlot3D[0<x<4-y^2&&0<z<2-y&&0<y<2,{x,-1,5},{y,0,2},{z,0,2},AxesLabel->{"X","Y","Z"},PlotPoints->50,PlotRange->All]

Note 1:  If you use too few PlotPoints (or use its default value), the surfaces will appear as very rough and will not be recognizable. On the other hand, if you use too many PlotPoints, the object will take a long time to render. 

Note 2:  The ranges for  x, y, z  still need to be supplied as part of the syntax, even though limits for the variables are already stated in the command. Make sure that the ranges for the variables are wider than the stated limits. For example, above, the lower limit in the range of  x  (i.e., -1)  is lower than the lower limit for x  stated in the command (i.e., 0).  On the other hand, the plot below shows what happens is the range limits are too restrictive: we have set the lower limit in the range of  x  to  2.

RegionPlot3D[0<x<4-y^2&&0<z<2-y&&0<y<2,{x,2,5},{y,0,2},{z,0,2},AxesLabel->{"X","Y","Z"},PlotPoints->50,PlotRange->All]