Math 151-007 Lab 12/5/07 Michael Pelsmajer and Oscar Ortega Maple can do nice 3D plots, and it's particularly easy to do volumes of revolution. So you can double check your hand-drawn sketches, and develop correct spatial intuition. Start by running these two commands: restart; with(Student[Calculus1]): Consider the region bounded by 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and 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 with 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 f := x -> sin(x); g := x -> (1/8)*x^2; plot({f,g}, 0..2, scaling=constrained); JSFH (It's the region between the two curves.) Let's rotate it about the x-axis. Often we'd like to (1) graph it, (2) give an integral that finds the area, and (3) compute the area. No problem: VolumeOfRevolution(f(x),g(x),x=0..2, output=plot, axis=horizontal); Before continuing, click the left mouse button on the plot, and while holding the mouse button down, move the mouse around. You should see the object move around. VolumeOfRevolution(f(x),g(x),x=0..2, output=integral, axis=horizontal); VolumeOfRevolution(f(x),g(x),x=0..2, output=value, axis=horizontal); JSFH Great! Well, I think so, at least, but actually that integral looks a little weird to me, and the plot could be better, (e.g. we couldn't see the inside until it was rotated manually). 1.a. For the integral, should we slice to get "washers" or "shells"? 1.b. What is the volume of one of these thin slices? 1.c. What integral do you get? I expect it will be not exactly what Maple says. Note: I'm not asking you to actually evaluate the integral, just to set it up. (You'll learn how to do this integral in Calculus II.) 1.d. Explain why your integral and Maple's integral will give the same result. Note: This should be obvious; you shouldn't have to evaluate the integral. Next, we improve the plot. I will make the outer surface rather transparent, and I'll add a grid so I can see the curvature better. I'll change the appearance of the inner surface too, just because I feel like it. VolumeOfRevolution(f(x),g(x),x=0..2, output=plot, axis=horizontal, volumeoptions=[transparency=.4, style=surfacewireframe], volume2options=[style=wireframeopaque, color=gold]); Alternatively, this is also nice, and typical : VolumeOfRevolution(f(x),g(x),x=0..2, output=plot, axis=horizontal, volumeoptions=[style=surfacewireframe, shading=xyz], volume2options=[style=surfacewireframe, shading=xyz]); Be sure to left-click on the two previous ones and move them around, and look at the Maple code to see how the effects are produced. We'd like to be able to rotate about any line. What about LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIjFGJ0Y5Rjk= or LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1GNjYtUSomdW1pbnVzMDtGJ0Y5RjtGPkZARkJGREZGRkgvRktRLDAuMjIyMjIyMmVtRicvRk5GUy1JI21uR0YkNiRRIjJGJ0Y5Rjk=? (The x-axis is LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIjBGJ0Y5Rjk=.) VolumeOfRevolution(f(x),g(x),x=0..2, output=plot, axis=horizontal, distancefromaxis=1); VolumeOfRevolution(f(x),g(x),x=0..2, output=plot, axis=horizontal, distancefromaxis=-2); JSFH 2.a. Modify the previous two plots so that (1) the right side doesn't stop at the line y=2; instead it goes until the curves intersect again, and (2) the plots are easier to see. 2.b. Compute these two volumes. 3.a. Now rotate the original region about the y-axis. (Figure it out. Or make a good guess.) Plot it and compute the volume. 3.a. Now rotate the original region about the line LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1GNjYtUSomdW1pbnVzMDtGJ0Y5RjtGPkZARkJGREZGRkgvRktRLDAuMjIyMjIyMmVtRicvRk5GUy1JI21uR0YkNiRRIjJGJ0Y5Rjk=. No plot, just compute the volume. 4. Make a cool 3D plot. (Here, "cool" is nontechnical.)