Math 151-005 Lab 11/21/08 Michael Pelsmajer and Jiarui Yang restart; with(Student[Calculus1]): 1. Using Maple to get intuition about approximating area by rectangles (Riemann Sums) Running the following command will make a new window. ApproximateIntTutor(x*cos(x)); You should see a plot of a function, with rectangles representing a Riemann Sum. "Area (Approximate Integral)" is the Riemann Sum: the sum of the signed area of the rectangles. Whenever you make a modification, click "Display" (or, depending what you want to see, "Animate") to see the results. 1.a. Change the number of intervals to be 7. How did you do it? What was \316\224x before, and what is \316\224x now? 1.b. Change the sample points to be randomly chosen in their intervals. Click "Display" several times - what happens? In particular, how does the 4th interval change, and how must that be affecting the Riemann Sum? (A clear, precise answer will make use of " x4* ".) 1.c. Click "Animate". What's happens to the plot of the Riemann Sum, and what happens to the Riemann Sum itself? 2. Using Maple to get intuition about volumes of revolution Maple is good for plotting 3-dimensional objects. Volumes of revolution are particularly easy to produce. Here are a couple curves, for an example. We are interested in (1) the area of the region between the curves, and (2) the volume that results from rotating that region about the x-axis (or about another line). f := x -> sin(x); g := x -> (1/8)*x^2; plot({f,g}, 0..2, scaling=constrained);

Running the following command will make a new window, that shows the volume obtained by rotating the region about the x-axis. VolumeOfRevolutionTutor(sin(x),x^2/8); Click on "Plot Options", check "Use Contstrained Scaling", and click OK. (This way, it won't stretch or squish the picture to better fit in a square display window. It's not a big deal - just do it, please.) Now, left-click on the plot, and drag the mouse to rotate the object, so that you can see what the green surface inside looks like. Every time you make a modification, you should click the "Display" or "Animate" button and then left-click/drag to rotate the object so you can see it from various angles! 2.a. Under "Line of Revolution", change the number to -1. What is the effect? 2.b. Now change it to 1.1. Note that the max of x*cos*(x) on that interval is 1, just a bit less than 1.1. How does that affect the current plot? 2.c. Now click on the Vertical bubble and change the number to 0. What's the line of revolution now? 2.d. Click the Horizontal bubble, and change g(x) to be (1/5)*x^2. Now switch the Display bubble from Volume to Disks. (And Display, etc.) Then switch it to Both. (And Display, etc.) Then click the Animate button. Describe what you see. (If you switch back to "Vertical", it will show you "shells" instead of disks/washers.) 3. Volumes with more plot options You can modify colors, make surfaces transparent, add/remove wireframes, etc., to make it easier to see the surface better. Here are a couple examples that shows what you can do. After running the commands, left-click and drag to rotate so that you can see the objects from various angles! VolumeOfRevolution(f(x),g(x),x=0..2, output=plot, axis=horizontal, volumeoptions=[transparency=.4, style=surfacewireframe], volume2options=[style=wireframeopaque, color="Gold"]); VolumeOfRevolution(f(x),g(x),x=0..2, output=plot, axis=horizontal, volumeoptions=[style=surfacewireframe, shading=xyz], volume2options=[style=surface, shading=xyz]); VolumeOfRevolution(f(x),g(x),x=0.., output=plot, axis=horizontal, distancefromaxis=1.1, volumeoptions=[style=surfacewireframe, shading=xyz], volume2options=[style=surface, shading=xyz]); You can also express the volume as an integral (not necessarily the same way that you've been taught, with respect to shells vs. washers), and compute the volume. VolumeOfRevolution(f(x),g(x),x=0..2, output=integral, axis=horizontal); VolumeOfRevolution(f(x),g(x),x=0..2, output=value, axis=horizontal); 3.a. For the second and third plots, the only change is the addition of distancefromaxis=1.1, so why it smooth on the outside and wireframe on the inside now? 3.b. Make a plot of the volume we get from: - the region between f and g from x = 0 to 2.3 - rotated about the axis y = LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbW9HRiQ2LVEqJnVtaW51czA7RicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjIyMjIyMmVtRicvJSdyc3BhY2VHRkMtSSNtbkdGJDYkUSIyRidGL0Yv - both surfaces should be partially transparent - the surface obtained by rotating f should have a "Turquoise" wireframe - the other surface should have that rainbow shading, and no wireframe.