2. 7.3a: Volumes Learning Goals ©2007 Roy L. Gover (www.mrgover.com) Use integration to calculate volumes of solids using the Disk and Washer Methods. Use integration to calculate volumes of solids with known cross sections

3. Important Idea We know formulas that will calculate the volume of cylinders, prisms, cones and spheres. These formulas were developed using Calculus. Calculus can be used to calculate volumes of any solid shape.

4. Examples Created by Lawrence Hunsh

5. Example from [0,2] Rotate about x axis Creates Partition with disks

6. Pick One Disk Important Idea Volume of each disk=

7. Important Idea Volume of each disk= Sum an infinite number of disks each infinitely thin

8. The total volume is given by the Riemann Sum: The definite integral accumulates the disk volumes Important Idea

9. Example Find the volume of the solid of revolution formed by revolving the graph: around the x axis. from [0,2] 0 2 r dx

10. Find the volume of the solid of revolution formed by revolving the graph: around the x axis. R is a constant R-R r dx Example

11. Definition To find volume of a solid of revolution use one of the following: Hori. Axis of Rev. Vert. Axis of Rev. Disk Method:

12. Example Find the volume of the solid of revolution formed by revolving the graph: around the x axis. R is a constant

13. Try This Find the volume of the solid formed by revolving the graph around the x axis. 0  cu. units

14. Try This Find the volume of the solid formed by revolving the graph around thefrom x=0 to x=1 x axis.Show your integral setup and evaluate to 3 decimal places with your calculator.

15. Solution 0 1Setup Top half of Disks

16. Example Find the volume of the solid of revolution formed by revolving the graph around the y axis. 0 1

17. Try This Find the volume of the solid of revolution formed by revolving the 0 2 region bounded by, & x= 2 about the y axis.

18. Example Find volume of the solid formed by revolving the region bounded by the graphs y= 0, x= & about the line x=. 8 02

19. Try This Find the volume of the solid formed by revolving the region bounded by the graphs y=x 2, y= 0 & x= 3 about the line x= 3. 0 3

20. Definition If the disk has a hole, it is a washer R r Vol. Of washer Washer Method

21. Important Idea The volume of a solid of revolution with a hole is the volume of the solid without the hole less the volume of the hole.

22. Example Find the volume of the solid formed by revolving the region bounded by the graphs y=x and y=x 2 about the x axis. R r

23. Example Find the volume of the solid formed by revolving the region bounded by the graphs y=x and y=x 2 about the y axis. R r

24. Try This Find the volume of the solid generated when the region between the graphs f(x)= over the interval [0,2] is revolved about the x axis. and g(x)=x

25. Example Find the volume of the solid formed by revolving the region bounded by the graphs y=2x 2, y= 0 & x = 2 about the line y=8. 8 0 2

26. Try This Find the volume of the solid formed by revolving the region bounded by the graphs, y= 0 & x = 1, x = 4, about the line y=4. 1 4 0 4

27. Solution 1 4 0 4

28. Definition Known Cross Sections Method Volume can be calculated by finding area of known geometric shapes and multiplying by thickness ( dx )

29. Important Idea Disks and washers are special cases of the known cross sections method where the cross sections are circles. The cross- sections can just as well be rectangles, triangles or semi-circles.

30. Important Idea We can find the area,, of each cross section, then add an infinite number of infinitely thin cross sections. When we multiply by thickness, we have volume. Volume=

31. Mathematica Examples Cross sections may be rectangles, semi-circles or triangles. The base of the solid may be a rectangle,circle, triangle or an irregular shape.

32. Example Find the volume of the solid whose base is bounded by the circle x 2 +y 2 =4 with equilateral triangle cross sections perpendicular to the x axis. x y

33. Try This Find the volume of the solid whose base is bounded by the circle x 2 +y 2 =4 with semicircular cross sections perpendicular to the x axis. x y

34. Example Find the volume of the solid formed with the region defined by and as the base and cross sections that are squares perpendicular to the base and the x -axis.

35. Lesson Close Write a paragraph describing how you find the volume of a solid of revolution.

36. Practice 1. 406/7,10,11-14,29(a) (slides 1-12 disks about x axis) 2. 406/8,9,23,24,29(b) (slides 13-14 disks about y-axis) 3. 406/29(c),31(a) (slides 15-16 disks about a line) 4. 406/15- 20,27,28,29(d)31(b) (slides 17-24 washers) 5. 406/39-42 (slides 25-32 cross sections)

37. “Solids of Known Cross Sections” from The Wolfram Demonstrations Project http://demonstrations.wolfram.com/SolidsOfKnownCrossSection/ Credits Dr. Lou Talman, Metro State University, Denver, CO. "Solids of Revolution." [Online image] 29 December 2004.. Unknown Author,"Solids of Revolution." [Online image] December 2004. Lawrence S. Husch, University of Tenn., Knoxville. "Visual Calculus-Solids of Revolution." [Online image] 29 December 2004..