Volumes of Revolution
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1 Mathematics Volumes of Revolution About this Lesson This lesson provides students with a physical method to visualize -dimensional solids and a specific procedure to sketch a solid of revolution. Students determine the perimeter and area of two-dimensional figures created by graphing equations on a coordinate plane. In addition, students determine the volume and surface area of three-dimensional figures created by revolving a planar region about a horizontal or vertical line. Prior to the lesson, students should be able to graph equations on a coordinate plane, calculate areas of plane figures, and calculate volumes of solids. This lesson is included in Module Areas and Volumes. Objectives Students will graph equations that define a plane figure on a coordinate plane. determine the perimeter and area of that plane figure. draw and describe the solid formed by revolving the plane figure about a vertical or horizontal line. calculate the volume and surface area of the solid. compare the volumes determined when revolving about different axes. Level Geometry Common Core State Standards for Mathematical Content This lesson addresses the following Common Core State Standards for Mathematical Content. The lesson requires that students recall and apply each of these standards rather than providing the initial introduction to the specific skill. The star symbol ( ) at the end of a specific standard indicates that the high school standard is connected to modeling. Explicitly addressed in this lesson Code Standard Level of Thinking G-GMD. Identify the shapes of two-dimensional crosssections Analyze of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects. G-GMD. G-GPE.7 Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. Analyze Analyze Depth of Knowledge III III III Copyright 0 Laying the Foundation, Inc., Dallas, TX. All rights reserved. Visit us online at i
2 Code Standard Level of Thinking F-IF.7a Graph functions expressed symbolically and Apply show key features of the graph, by hand in simple cases and using technology for more complicated cases. Graph linear and quadratic functions and show intercepts, maxima, and minima. A-REI. Graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes. Teacher Overview Volumes of Revolution Analyze Depth of Knowledge II III Common Core State Standards for Mathematical Practice These standards describe a variety of instructional practices based on processes and proficiencies that are critical for mathematics instruction. LTF incorporates these important processes and proficiencies to help students develop knowledge and understanding and to assist them in making important connections across grade levels. This lesson allows teachers to address the following Common Core State Standards for Mathematical Practice. Implicitly addressed in this lesson Code Standard Make sense of problems and persevere in solving them. Reason abstractly and quantitatively. Construct viable arguments and critique the reasoning of others. Model with mathematics. 6 Attend to precision. Copyright 0 Laying the Foundation, Inc., Dallas, TX. All rights reserved. Visit us online at ii
3 Teacher Overview Volumes of Revolution LTF Content Progression Chart In the spirit of LTF s goal to connect mathematics across grade levels, the Content Progression Chart demonstrates how specific skills build and develop from sixth grade through pre-calculus. Each column, under a grade level or course heading, lists the concepts and skills that students in that grade or course should master. Each row illustrates how a specific skill is developed as students advance through their mathematics courses. 6th Grade Skills/Objectives Given or coordinate points that form a triangle or a rectangle with horizontal line and vertical line, calculate the area of the figure. (00_06.AV_B0) Given or coordinate points that form a triangle or a rectangle with horizontal or vertical line, calculate the surface area and/or volume of the cone or cylinder formed by revolving the bounded region about either of the lines. (00_06.AV_B0) 7th Grade Skills/Objectives Given or coordinate points that form a triangle or a rectangle with horizontal line and vertical line, calculate the area of the figure. (00_07.AV_B0) Given or coordinate points that form a triangle or a rectangle with horizontal or vertical line, calculate the surface area and/or volume of the cone or cylinder formed by revolving the bounded region about either of the lines. (00_07.AV_B0) Algebra Skills/Objectives Calculate the area of a triangle, rectangle, trapezoid, or composite of these three figures formed by linear equations and/or determine the equations of the lines that bound the figure. (00_A.AV_B0) Given or coordinate points that form a triangle or a rectangle with horizontal or vertical line, calculate the surface area and/or volume of the cone or cylinder formed by revolving the bounded region about either of the lines. (00_07.AV_B0) Geometry Skills/Objectives Calculate the area of a triangle, rectangle, trapezoid, circle, or composite of these figures formed by linear equations or Algebra Skills/Objectives Calculate the area of a triangle, rectangle, trapezoid, circle, or composite of these figures formed by linear equations, linear equations, equations of circles linear inequalities, or linear inequalities, or and/or determine the conic equations conic equations equations of the lines and/or determine the and/or determine the and circles that equations of the lines bound the figure. (00_GE.AV_B0) Given the equations of lines or circles that bound a triangular, rectangular, trapezoidal, or circular region, calculate the volume and/or surface area of the region formed by revolving the region about a horizontal or vertical line. (00_GE.AV_B0) equations of the lines and circles that bound the figure. (00_A.AV_B0) Given the equations of lines or circles or a set of inequalities that bound a triangular, rectangular, trapezoidal, or circular region, calculate the volume and/or surface area of the region formed by revolving the region about a horizontal or vertical line. (00_A.AV_B0) Pre-Calculus Skills/Objectives Calculate the area of a triangle, rectangle, trapezoid, circle, or composite of these figures formed by and circles that bound the figure. (00_PC.AV_B0) Given the equations of lines or circles or a set of inequalities that bound a triangular, rectangular, trapezoidal, or circular region, calculate the volume and/or surface area of the region formed by revolving the region about a horizontal or vertical line. (00_PC.AV_B0) Connection to AP* AP Calculus Topic: Areas and Volumes *Advanced Placement and AP are registered trademarks of the College Entrance Examination Board. The College Board was not involved in the production of this product. Materials and Resources Student Activity pages Cut-out of planar region Pencil, straw, or skewer Tape Straight edge Applets for creating volumes of revolution: Copyright 0 Laying the Foundation, Inc., Dallas, TX. All rights reserved. Visit us online at iii
4 Assessments The following types of formative assessments are embedded in this lesson: Students engage in independent practice. Students construct and use manipulatives. The following additional assessments are located on the LTF website: Areas and Volumes Geometry Free Response Questions Areas and Volumes Geometry Multiple Choice Questions Teacher Overview Volumes of Revolution Teaching Suggestions The concept of revolving a region about an axis is fundamental to integral calculus. This lesson includes calculating perimeter and area of a planar region and then determining the volume and surface area generated by revolving the region around the x-axis or the y-axis. Make sure that students are correctly identifying the radius and the height for each generated solid. Ask students to explain how changing the axis of revolution changes the dimensions of the solid that is formed. To help students visualize the solid generated by revolving the figure about an axis, have them glue or tape a triangle onto a pencil, straw or skewer and then revolve the triangle horizontally and vertically by rolling the stick between their hands so the students can see the cone that will be generated. An extension of this lesson is to bring three-dimensional objects to class and have the students draw cross-sections. Being able to visualize solids and their cross sections is a valuable tool for calculus. When drawing a sketch of a solid revolution, use the following procedure: Draw the boundaries. Shade the region to be revolved. Draw the reflection (mirror image) of the region about the axis of revolution. Connect significant points and their reflections with ellipses (oval shapes) to illustrate the solid nature of the figure. Modality LTF emphasizes using multiple representations to connect various approaches to a situation in order to increase student understanding. The lesson provides multiple strategies and models for using these representations to introduce, explore, and reinforce mathematical concepts and to enhance conceptual understanding. P Physical V Verbal A Analytical N Numerical G Graphical Copyright 0 Laying the Foundation, Inc., Dallas, TX. All rights reserved. Visit us online at iv
5 Answers. a. Teacher Overview Volumes of Revolution b. units c. 0 sq. units d. e. a cylinder with a cylinder removed f. V R h r h h R r () 0 cu. units Copyright 0 Laying the Foundation, Inc., Dallas, TX. All rights reserved. Visit us online at v
6 . a. Teacher Overview Volumes of Revolution b. units c. 6 sq. units d e. cone f. cu. units g. sq. units (remember the base) h. The volume is 6 cu. units, so the y-axis rotation has greater volume. The surface area, 6 sq. units, is greater in the y-axis rotation as well. i. Region bounded by y x, y 0 and x 0 Copyright 0 Laying the Foundation, Inc., Dallas, TX. All rights reserved. Visit us online at vi
7 . a. Teacher Overview Volumes of Revolution b. ( +) units c. sq. units d e. sphere f. cu. units g. 6 sq. units h. The geometric solid would be the top half of the sphere, called a hemisphere. The volume would be half the volume of the sphere, or 6 cu. units. Copyright 0 Laying the Foundation, Inc., Dallas, TX. All rights reserved. Visit us online at vii
8 . a. Teacher Overview Volumes of Revolution b. c. Copyright 0 Laying the Foundation, Inc., Dallas, TX. All rights reserved. Visit us online at viii
9 Mathematics Volumes of Revolution. Sketch the region bounded by the lines y =, y =, x =, x = 6. a b. Determine the perimeter of the region. c. Determine the area of the region. d. Draw a picture of the region being revolved about the x-axis e. Describe the geometric solid formed by revolving the region about the x-axis. f. Determine the volume of the geometric solid. Copyright 0 Laying the Foundation, Inc., Dallas, TX. All rights reserved. Visit us online at
10 . Sketch the region bounded by the lines y = x, y = 0, x = 0. Student Activity Volumes of Revolution a b. Determine the perimeter of the region. c. Determine the area of the region. d. Draw a picture of the region being revolved about the x-axis e. What geometric figure is formed by revolving the region about the x-axis? f. Determine the volume of the geometric solid. g. Determine the surface area of the geometric solid. h. If the region were revolved about the y-axis, would the volume be greater than, less than, or equal to the volume formed by revolving about the x-axis? Justify your answer. Compare the surface areas. i. Name another region that could be revolved about the x-axis to create exactly the same geometric solid. Copyright 0 Laying the Foundation, Inc., Dallas, TX. All rights reserved. Visit us online at
11 Student Activity Volumes of Revolution. Sketch the region bounded by the curve y = x and the line y = 0. a b. Determine the perimeter of the region. c. Determine the area of the region. d. Draw a picture of the region being revolved about the x-axis e. What geometric figure is formed by revolving the region about the x-axis? f. Determine the volume of the geometric solid. g. Determine the surface area of the geometric solid. h. If the region were revolved about the y-axis, would the volume be greater than, less than, or equal to the volume formed by revolving about the x-axis? Justify your answer. Copyright 0 Laying the Foundation, Inc., Dallas, TX. All rights reserved. Visit us online at
12 Student Activity Volumes of Revolution. A region is bounded by the graphs of y ( x ) and y x. a. Draw a picture of the region. b. Draw a picture of the region revolved around the x-axis. y x y x c. Draw a picture of the region revolved around the y-axis. y x When drawing a sketch of a solid revolution, use the following procedure: Draw the boundaries. Shade the region to be revolved. Draw the reflection (mirror image) of the region across the axis of revolution. Connect significant points and their reflections with ellipses (oval shapes) to illustrate the solid nature of the figure. Copyright 0 Laying the Foundation, Inc., Dallas, TX. All rights reserved. Visit us online at
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