Quote of the Day: "One can invent mathematics without knowing much of its history. One can use mathematics without knowing much, if any, of its history. But one cannot have a mature appreciation of mathematics without a substantial knowledge of its history." -- Abe Shenitzer Objectives: The student will compute the volume of solids of revolution using the disk method (slicing). Materials needed: Styrofoam disks of various sizes along with a dowel. Power Drill and safety glasses. Styrofoam cut-outs mounted on dowels. 1. Collect homework. 2. Recall that to find the area of a plane region, we divide the region into thin rectangles, add the areas of rectangles to form a Riemann sum, and then take the limit of the Riemann sums to obtain an integral for the area: 3. This week, we are going to examine volumes of solids of revolution. We use the same strategy to find the volume of a solid. We will divide the solid into thin slabs, approximate the volume of each slab, add the approximations together to form a Riemann sum, and then take the limit of the Riemann sums to form an integral for the volume of the solid. In our first method, we will be summing up volumes of disks (or cylinders).
Show Styrofoam circles on a dowel to illustrate this idea. 4. Example
5. One of the most difficult things to do when working with volumes of solids of revolution is to visualize the shape that is being formed. To help with this visualization process, use one of the following techniques: (A) Use styrofoam disks Click here for a picture (B) Use power drill with Styrofoam cut-outs mounted on dowels. When the styrofoam rotates it traces out the solid of revolution. Click here for a picture (C) Do some edible calculus (by Nancy Dirnberger): (USE after SHELL METHOD) If you core an apple you have a great solid of revolution with a hole through the solid! An apple is usually sliced in one of two ways. If you slice it so that the plane of the slice contains what would be the axis of revolution, your apple slice (actually you have two) is a disc of revolution. You can almost slice it thin enough to have width dx! If you slice the apple into rings, the resulting slices are washers that will give you the same volume when added. With a large, thick slice of a Bermuda onion you can pull up onion rings as successive cylindrical shells. Students often have a problem visualizing these cylindrical shells. 6. Determine the volume of a sphere. 7. Example (with respect to the y-axis) 8. Assignment: p. 456 (1, 3, 7, 10, 13) Worksheet "Dorothy and Kansas" Problem (PDF File) Worksheet "Dorothy and Kansas" Problem (Word DOCUMENT)
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2014 3D Geometry Project Description and Rubric
After the AP Exam, 'C' Topics Project [doc] [pdf]
Laura Veuve, Leadership High School, San Francisco: I put several teachers' ideas and worksheets together for a "C Topics" project where students in pairs or small teams choose a topic from a list of "C topics" (as opposed to AB topics) and then teach it to the class, with a homework assignment and 3 test questions. I'd be happy to share the assignment explanation, expectations, and rubric with you all - I can only take credit for writing a little of it, but I like the way it all came together and I think it's useful. You'll need to adjust it if you have larger classes or a different end-of-year schedule, of course. And the page numbers are from FDWK.
How Sweet It Is: Volumes of Revolution with Candy [doc] [pdf]
Dixie Ross, Pflugerville HS, Texas: The idea is to have the kids sketch each region and then ask them “what does this look like?” They will say various things and when they finally hit Egg! for the first one you pull out the chocolate Easter eggs and pass them out along with plastic knives. Have them hold egg horizontally and think about slicing it with the knife. Should we slice with respect to x or y? They will decide with respect to x and you set up the problem showing how the thickness of the slice is dx. Tell them they can “dispose of their volume model” once they have worked the problem correctly. Materials: Chocolate eggs, Hershey’s kisses, Reese’s peanut butter cups, gumdrops, Vanilla Wafer cookies, golf tees
Video Analysis of Speed Scenarios: projects
Using VideoPoint software, students analyze the position, velocity, acceleration and speed of a hovercraft under different scenarios.
Volumes of Known Cross Sections
Here's a worksheet and directions for making models of volumes with cross sections of
isosceles right triangles
Calculating the Volume of a Vase
Volumes of Revolution
My students have always had difficulty visualizing volumes formed by revolving a bounded area around an axis. I made some plywood areas, attached them to dowels and spun them with an electric drill. The radius of the disk or washer is represented as a white rectangle.
|Disks: These are the most basic forms used.|
|These are rotated about the y or the x-axis by changing the orientation of the drill.|
|The same area is a disk or a washer depending on the axis of rotation.|
Washer method: the same area is rotated around the y or the x-axis.